diff --git a/src/array_api_extra/_lib/_at.py b/src/array_api_extra/_lib/_at.py
index 22e18d2..736f77b 100644
--- a/src/array_api_extra/_lib/_at.py
+++ b/src/array_api_extra/_lib/_at.py
@@ -344,6 +344,7 @@ def _op(
             msg = f"Can't update read-only array {x}"
             raise ValueError(msg)
 
+        # Backends without boolean indexing (other than JAX) crash here
         if in_place_op:  # add(), subtract(), ...
             x[idx] = in_place_op(x[idx], y)
         else:  # set()
diff --git a/src/array_api_extra/_lib/_backends.py b/src/array_api_extra/_lib/_backends.py
index f044281..3beb676 100644
--- a/src/array_api_extra/_lib/_backends.py
+++ b/src/array_api_extra/_lib/_backends.py
@@ -1,9 +1,10 @@
 """Backends with which array-api-extra interacts in delegation and testing."""
 
+from __future__ import annotations
+
 from collections.abc import Callable
 from enum import Enum
 from types import ModuleType
-from typing import cast
 
 from ._utils import _compat
 
@@ -23,9 +24,14 @@ class Backend(Enum):  # numpydoc ignore=PR01,PR02  # type: ignore[no-subclass-an
         corresponding to the backend.
     """
 
+    # Use :<tag> to prevent Enum from deduplicating items with the same value
     ARRAY_API_STRICT = "array_api_strict", _compat.is_array_api_strict_namespace
+    ARRAY_API_STRICTEST = (
+        "array_api_strict:strictest",
+        _compat.is_array_api_strict_namespace,
+    )
     NUMPY = "numpy", _compat.is_numpy_namespace
-    NUMPY_READONLY = "numpy_readonly", _compat.is_numpy_namespace
+    NUMPY_READONLY = "numpy:readonly", _compat.is_numpy_namespace
     CUPY = "cupy", _compat.is_cupy_namespace
     TORCH = "torch", _compat.is_torch_namespace
     DASK = "dask.array", _compat.is_dask_namespace
@@ -48,4 +54,13 @@ def __init__(
 
     def __str__(self) -> str:  # type: ignore[explicit-override]  # pyright: ignore[reportImplicitOverride]  # numpydoc ignore=RT01
         """Pretty-print parameterized test names."""
-        return cast(str, self.value)
+        return self.name.lower()
+
+    @property
+    def modname(self) -> str:  # numpydoc ignore=RT01
+        """Module name to be imported."""
+        return self.value.split(":")[0]
+
+    def like(self, *others: Backend) -> bool:  # numpydoc ignore=PR01,RT01
+        """Check if this backend uses the same module as others."""
+        return any(self.modname == other.modname for other in others)
diff --git a/src/array_api_extra/_lib/_funcs.py b/src/array_api_extra/_lib/_funcs.py
index efe2f37..e552392 100644
--- a/src/array_api_extra/_lib/_funcs.py
+++ b/src/array_api_extra/_lib/_funcs.py
@@ -8,13 +8,14 @@
 
 from ._at import at
 from ._utils import _compat, _helpers
-from ._utils._compat import (
-    array_namespace,
-    is_dask_namespace,
-    is_jax_array,
-    is_jax_namespace,
+from ._utils._compat import array_namespace, is_dask_namespace, is_jax_array
+from ._utils._helpers import (
+    asarrays,
+    capabilities,
+    eager_shape,
+    meta_namespace,
+    ndindex,
 )
-from ._utils._helpers import asarrays, eager_shape, meta_namespace, ndindex
 from ._utils._typing import Array
 
 __all__ = [
@@ -152,7 +153,7 @@ def _apply_where(  # type: ignore[explicit-any]  # numpydoc ignore=PR01,RT01
 ) -> Array:
     """Helper of `apply_where`. On Dask, this runs on a single chunk."""
 
-    if is_jax_namespace(xp):
+    if not capabilities(xp)["boolean indexing"]:
         # jax.jit does not support assignment by boolean mask
         return xp.where(cond, f1(*args), f2(*args) if f2 is not None else fill_value)
 
@@ -708,14 +709,34 @@ def nunique(x: Array, /, *, xp: ModuleType | None = None) -> Array:
         # size= is JAX-specific
         # https://github.com/data-apis/array-api/issues/883
         _, counts = xp.unique_counts(x, size=_compat.size(x))
-        return xp.astype(counts, xp.bool).sum()
-
-    _, counts = xp.unique_counts(x)
-    n = _compat.size(counts)
-    # FIXME https://github.com/data-apis/array-api-compat/pull/231
-    if n is None:  # e.g. Dask, ndonnx
-        return xp.astype(counts, xp.bool).sum()
-    return xp.asarray(n, device=_compat.device(x))
+        return (counts > 0).sum()
+
+    # There are 3 general use cases:
+    # 1. backend has unique_counts and it returns an array with known shape
+    # 2. backend has unique_counts and it returns a None-sized array;
+    #    e.g. Dask, ndonnx
+    # 3. backend does not have unique_counts; e.g. wrapped JAX
+    if capabilities(xp)["data-dependent shapes"]:
+        # xp has unique_counts; O(n) complexity
+        _, counts = xp.unique_counts(x)
+        n = _compat.size(counts)
+        if n is None:
+            return xp.sum(xp.ones_like(counts))
+        return xp.asarray(n, device=_compat.device(x))
+
+    # xp does not have unique_counts; O(n*logn) complexity
+    x = xp.sort(xp.reshape(x, -1))
+    mask = x != xp.roll(x, -1)
+    default_int = xp.__array_namespace_info__().default_dtypes(
+        device=_compat.device(x)
+    )["integral"]
+    return xp.maximum(
+        # Special cases:
+        # - array is size 0
+        # - array has all elements equal to each other
+        xp.astype(xp.any(~mask), default_int),
+        xp.sum(xp.astype(mask, default_int)),
+    )
 
 
 def pad(
diff --git a/src/array_api_extra/_lib/_utils/_helpers.py b/src/array_api_extra/_lib/_utils/_helpers.py
index 9882d72..6400627 100644
--- a/src/array_api_extra/_lib/_utils/_helpers.py
+++ b/src/array_api_extra/_lib/_utils/_helpers.py
@@ -12,7 +12,9 @@
     array_namespace,
     is_array_api_obj,
     is_dask_namespace,
+    is_jax_namespace,
     is_numpy_array,
+    is_pydata_sparse_namespace,
 )
 from ._typing import Array
 
@@ -23,6 +25,7 @@
 
 __all__ = [
     "asarrays",
+    "capabilities",
     "eager_shape",
     "in1d",
     "is_python_scalar",
@@ -270,3 +273,36 @@ def meta_namespace(
     # Quietly skip scalars and None's
     metas = [cast(Array | None, getattr(a, "_meta", None)) for a in arrays]
     return array_namespace(*metas)
+
+
+def capabilities(xp: ModuleType) -> dict[str, int]:
+    """
+    Return patched ``xp.__array_namespace_info__().capabilities()``.
+
+    TODO this helper should be eventually removed once all the special cases
+    it handles are fixed in the respective backends.
+
+    Parameters
+    ----------
+    xp : array_namespace
+        The standard-compatible namespace.
+
+    Returns
+    -------
+    dict
+        Capabilities of the namespace.
+    """
+    if is_pydata_sparse_namespace(xp):
+        # No __array_namespace_info__(); no indexing by sparse arrays
+        return {"boolean indexing": False, "data-dependent shapes": True}
+    out = xp.__array_namespace_info__().capabilities()
+    if is_jax_namespace(xp):
+        # FIXME https://github.com/jax-ml/jax/issues/27418
+        out = out.copy()
+        out["boolean indexing"] = False
+    if is_dask_namespace(xp):
+        # FIXME https://github.com/data-apis/array-api-compat/pull/290
+        out = out.copy()
+        out["boolean indexing"] = True
+        out["data-dependent shapes"] = True
+    return out
diff --git a/tests/conftest.py b/tests/conftest.py
index 54e2a23..4e36885 100644
--- a/tests/conftest.py
+++ b/tests/conftest.py
@@ -1,6 +1,6 @@
 """Pytest fixtures."""
 
-from collections.abc import Callable
+from collections.abc import Callable, Generator
 from contextlib import suppress
 from functools import partial, wraps
 from types import ModuleType
@@ -19,6 +19,7 @@
 T = TypeVar("T")
 P = ParamSpec("P")
 
+NUMPY_VERSION = tuple(int(v) for v in np.__version__.split(".")[2])
 np_compat = array_namespace(np.empty(0))  # type: ignore[arg-type]  # pyright: ignore[reportArgumentType]
 
 
@@ -43,7 +44,7 @@ def library(request: pytest.FixtureRequest) -> Backend:  # numpydoc ignore=PR01,
                 msg = f"argument of {marker_name} must be a Backend enum"
                 raise TypeError(msg)
             if library == elem:
-                reason = library.value
+                reason = str(library)
                 with suppress(KeyError):
                     reason += ":" + cast(str, marker.kwargs["reason"])
                 skip_or_xfail(reason=reason)
@@ -104,7 +105,7 @@ def wrapper(*args: P.args, **kwargs: P.kwargs) -> T:  # numpydoc ignore=GL08
 @pytest.fixture
 def xp(
     library: Backend, request: pytest.FixtureRequest, monkeypatch: pytest.MonkeyPatch
-) -> ModuleType:  # numpydoc ignore=PR01,RT03
+) -> Generator[ModuleType]:  # numpydoc ignore=PR01,RT03
     """
     Parameterized fixture that iterates on all libraries.
 
@@ -113,25 +114,38 @@ def xp(
     The current array namespace.
     """
     if library == Backend.NUMPY_READONLY:
-        return NumPyReadOnly()  # type: ignore[return-value]  # pyright: ignore[reportReturnType]
-    xp = pytest.importorskip(library.value)
+        yield NumPyReadOnly()  # type: ignore[misc]  # pyright: ignore[reportReturnType]
+        return
+
+    if library.like(Backend.ARRAY_API_STRICT) and NUMPY_VERSION < (1, 26):
+        pytest.skip("array_api_strict is untested on NumPy <1.26")
+
+    xp = pytest.importorskip(library.modname)
     # Possibly wrap module with array_api_compat
     xp = array_namespace(xp.empty(0))
 
+    if library == Backend.ARRAY_API_STRICTEST:
+        with xp.ArrayAPIStrictFlags(
+            boolean_indexing=False,
+            data_dependent_shapes=False,
+            # writeable=False,  # TODO implement in array-api-strict
+            # lazy=True,  # TODO implement in array-api-strict
+            enabled_extensions=(),
+        ):
+            yield xp
+        return
+
     # On Dask and JAX, monkey-patch all functions tagged by `lazy_xp_function`
     # in the global scope of the module containing the test function.
     patch_lazy_xp_functions(request, monkeypatch, xp=xp)
 
-    if library == Backend.ARRAY_API_STRICT and np.__version__ < "1.26":
-        pytest.skip("array_api_strict is untested on NumPy <1.26")
-
     if library == Backend.JAX:
         import jax
 
         # suppress unused-ignore to run mypy in -e lint as well as -e dev
         jax.config.update("jax_enable_x64", True)  # type: ignore[no-untyped-call,unused-ignore]
 
-    return xp
+    yield xp
 
 
 @pytest.fixture(params=[Backend.DASK])  # Can select the test with `pytest -k dask`
diff --git a/tests/test_at.py b/tests/test_at.py
index 218b05b..4bde5ce 100644
--- a/tests/test_at.py
+++ b/tests/test_at.py
@@ -20,7 +20,8 @@
 pytestmark = [
     pytest.mark.skip_xp_backend(
         Backend.SPARSE, reason="read-only backend without .at support"
-    )
+    ),
+    pytest.mark.skip_xp_backend(Backend.ARRAY_API_STRICTEST, reason="boolean indexing"),
 ]
 
 
@@ -256,7 +257,7 @@ def test_incompatible_dtype(
     elif library is Backend.DASK:
         z = at_op(x, idx, op, 1.1, copy=copy)
 
-    elif library is Backend.ARRAY_API_STRICT and op is not _AtOp.SET:
+    elif library.like(Backend.ARRAY_API_STRICT) and op is not _AtOp.SET:
         with pytest.raises(Exception, match=r"cast|promote|dtype"):
             _ = at_op(x, idx, op, 1.1, copy=copy)
 
diff --git a/tests/test_funcs.py b/tests/test_funcs.py
index 46591ed..48ad7b0 100644
--- a/tests/test_funcs.py
+++ b/tests/test_funcs.py
@@ -32,6 +32,8 @@
 from array_api_extra._lib._utils._typing import Array, Device
 from array_api_extra.testing import lazy_xp_function
 
+from .conftest import NUMPY_VERSION
+
 # some xp backends are untyped
 # mypy: disable-error-code=no-untyped-def
 
@@ -48,12 +50,6 @@
 lazy_xp_function(sinc, static_argnames="xp")
 
 
-NUMPY_GE2 = int(np.__version__.split(".")[0]) >= 2
-
-
-@pytest.mark.skip_xp_backend(
-    Backend.SPARSE, reason="read-only backend without .at support"
-)
 class TestApplyWhere:
     @staticmethod
     def f1(x: Array, y: Array | int = 10) -> Array:
@@ -153,6 +149,14 @@ def test_dont_overwrite_fill_value(self, xp: ModuleType):
         xp_assert_equal(actual, xp.asarray([100, 12]))
         xp_assert_equal(fill_value, xp.asarray([100, 200]))
 
+    @pytest.mark.skip_xp_backend(
+        Backend.ARRAY_API_STRICTEST,
+        reason="no boolean indexing -> run everywhere",
+    )
+    @pytest.mark.skip_xp_backend(
+        Backend.SPARSE,
+        reason="no indexing by sparse array -> run everywhere",
+    )
     def test_dont_run_on_false(self, xp: ModuleType):
         x = xp.asarray([1.0, 2.0, 0.0])
         y = xp.asarray([0.0, 3.0, 4.0])
@@ -192,6 +196,7 @@ def test_device(self, xp: ModuleType, device: Device):
         y = apply_where(x % 2 == 0, x, self.f1, fill_value=x)
         assert get_device(y) == device
 
+    @pytest.mark.skip_xp_backend(Backend.SPARSE, reason="no isdtype")
     @pytest.mark.filterwarnings("ignore::RuntimeWarning")  # overflows, etc.
     @hypothesis.settings(
         # The xp and library fixtures are not regenerated between hypothesis iterations
@@ -217,8 +222,8 @@ def test_hypothesis(  # type: ignore[explicit-any,decorated-any]
         library: Backend,
     ):
         if (
-            library in (Backend.NUMPY, Backend.NUMPY_READONLY)
-            and not NUMPY_GE2
+            library.like(Backend.NUMPY)
+            and NUMPY_VERSION < (2, 0)
             and dtype is np.float32
         ):
             pytest.xfail(reason="NumPy 1.x dtype promotion for scalars")
@@ -562,6 +567,9 @@ def test_xp(self, xp: ModuleType):
         assert y.shape == (1, 1, 1, 3)
 
 
+@pytest.mark.filterwarnings(  # array_api_strictest
+    "ignore:invalid value encountered:RuntimeWarning:array_api_strict"
+)
 @pytest.mark.xfail_xp_backend(Backend.SPARSE, reason="no isdtype")
 class TestIsClose:
     @pytest.mark.parametrize("swap", [False, True])
@@ -680,6 +688,7 @@ def test_bool_dtype(self, xp: ModuleType):
             isclose(xp.asarray(True), b, atol=1), xp.asarray([True, True, True])
         )
 
+    @pytest.mark.skip_xp_backend(Backend.ARRAY_API_STRICTEST, reason="unknown shape")
     def test_none_shape(self, xp: ModuleType):
         a = xp.asarray([1, 5, 0])
         b = xp.asarray([1, 4, 2])
@@ -687,6 +696,7 @@ def test_none_shape(self, xp: ModuleType):
         a = a[a < 5]
         xp_assert_equal(isclose(a, b), xp.asarray([True, False]))
 
+    @pytest.mark.skip_xp_backend(Backend.ARRAY_API_STRICTEST, reason="unknown shape")
     def test_none_shape_bool(self, xp: ModuleType):
         a = xp.asarray([True, True, False])
         b = xp.asarray([True, False, True])
@@ -819,8 +829,27 @@ def test_empty(self, xp: ModuleType):
         a = xp.asarray([])
         xp_assert_equal(nunique(a), xp.asarray(0))
 
-    def test_device(self, xp: ModuleType, device: Device):
-        a = xp.asarray(0.0, device=device)
+    def test_size1(self, xp: ModuleType):
+        a = xp.asarray([123])
+        xp_assert_equal(nunique(a), xp.asarray(1))
+
+    def test_all_equal(self, xp: ModuleType):
+        a = xp.asarray([123, 123, 123])
+        xp_assert_equal(nunique(a), xp.asarray(1))
+
+    @pytest.mark.xfail_xp_backend(Backend.DASK, reason="No equal_nan kwarg in unique")
+    @pytest.mark.xfail_xp_backend(Backend.SPARSE, reason="sparse#855")
+    def test_nan(self, xp: ModuleType, library: Backend):
+        if library.like(Backend.NUMPY) and NUMPY_VERSION < (1, 24):
+            pytest.xfail("NumPy <1.24 has no equal_nan kwarg in unique")
+
+        # Each NaN is counted separately
+        a = xp.asarray([xp.nan, 123.0, xp.nan])
+        xp_assert_equal(nunique(a), xp.asarray(3))
+
+    @pytest.mark.parametrize("size", [0, 1, 2])
+    def test_device(self, xp: ModuleType, device: Device, size: int):
+        a = xp.asarray([0.0] * size, device=device)
         assert get_device(nunique(a)) == device
 
     def test_xp(self, xp: ModuleType):
@@ -895,6 +924,7 @@ def test_sequence_of_tuples_width(self, xp: ModuleType):
 
 
 @pytest.mark.xfail_xp_backend(Backend.SPARSE, reason="no argsort")
+@pytest.mark.skip_xp_backend(Backend.ARRAY_API_STRICTEST, reason="no unique_values")
 class TestSetDiff1D:
     @pytest.mark.xfail_xp_backend(Backend.DASK, reason="NaN-shaped arrays")
     @pytest.mark.xfail_xp_backend(
diff --git a/tests/test_helpers.py b/tests/test_helpers.py
index 81b11d1..c7d271c 100644
--- a/tests/test_helpers.py
+++ b/tests/test_helpers.py
@@ -10,6 +10,7 @@
 from array_api_extra._lib._utils._compat import device as get_device
 from array_api_extra._lib._utils._helpers import (
     asarrays,
+    capabilities,
     eager_shape,
     in1d,
     meta_namespace,
@@ -27,6 +28,7 @@
 
 
 @pytest.mark.xfail_xp_backend(Backend.SPARSE, reason="no unique_inverse")
+@pytest.mark.skip_xp_backend(Backend.ARRAY_API_STRICTEST, reason="no unique_inverse")
 class TestIn1D:
     # cover both code paths
     @pytest.mark.parametrize(
@@ -161,7 +163,8 @@ def test_ndindex(shape: tuple[int, ...]):
     assert tuple(ndindex(*shape)) == tuple(np.ndindex(*shape))
 
 
-@pytest.mark.xfail_xp_backend(Backend.SPARSE, reason="index by sparse array")
+@pytest.mark.skip_xp_backend(Backend.SPARSE, reason="index by sparse array")
+@pytest.mark.skip_xp_backend(Backend.ARRAY_API_STRICTEST, reason="boolean indexing")
 def test_eager_shape(xp: ModuleType, library: Backend):
     a = xp.asarray([1, 2, 3])
     # Lazy arrays, like Dask, have an eager shape until you slice them with
@@ -194,3 +197,10 @@ def test_dask_metas(self, da: ModuleType):
     def test_xp(self, xp: ModuleType):
         args = None, xp.asarray(0), 1
         assert meta_namespace(*args, xp=xp) in (xp, np_compat)
+
+
+def test_capabilities(xp: ModuleType):
+    expect = {"boolean indexing", "data-dependent shapes"}
+    if xp.__array_api_version__ >= "2024.12":
+        expect.add("max dimensions")
+    assert capabilities(xp).keys() == expect
diff --git a/tests/test_lazy.py b/tests/test_lazy.py
index 7057af8..d360e50 100644
--- a/tests/test_lazy.py
+++ b/tests/test_lazy.py
@@ -1,3 +1,4 @@
+import contextlib
 from types import ModuleType
 from typing import cast
 
@@ -214,24 +215,20 @@ def test_lazy_apply_none_shape_in_args(xp: ModuleType, library: Backend):
     mxp = np if library is Backend.DASK else xp
     int_type = xp.asarray(0).dtype
 
+    ctx: contextlib.AbstractContextManager[object]
     if library is Backend.JAX:
-        # Single output
-        with pytest.raises(ValueError, match="Output shape must be fully known"):
-            _ = lazy_apply(mxp.unique_values, x, shape=(None,))
-
-        # Multi output
-        with pytest.raises(ValueError, match="Output shape must be fully known"):
-            _ = lazy_apply(
-                mxp.unique_counts,
-                x,
-                shape=((None,), (None,)),
-                dtype=(x.dtype, int_type),
-            )
+        ctx = pytest.raises(ValueError, match="Output shape must be fully known")
+    elif library is Backend.ARRAY_API_STRICTEST:
+        ctx = pytest.raises(RuntimeError, match="data-dependent shapes")
     else:
-        # Single output
+        ctx = contextlib.nullcontext()
+
+    # Single output
+    with ctx:
         values = lazy_apply(mxp.unique_values, x, shape=(None,))
         xp_assert_equal(values, xp.asarray([1, 2]))
 
+    with ctx:
         # Multi output
         values, counts = lazy_apply(
             mxp.unique_counts,
@@ -255,8 +252,9 @@ def f(x: Array) -> Array:
 lazy_xp_function(check_lazy_apply_none_shape_broadcast)
 
 
-@pytest.mark.xfail_xp_backend(Backend.SPARSE, reason="bool mask")
-@pytest.mark.xfail_xp_backend(Backend.JAX, reason="unknown shape")
+@pytest.mark.skip_xp_backend(Backend.SPARSE, reason="index by sparse array")
+@pytest.mark.skip_xp_backend(Backend.JAX, reason="boolean indexing")
+@pytest.mark.skip_xp_backend(Backend.ARRAY_API_STRICTEST, reason="boolean indexing")
 def test_lazy_apply_none_shape_broadcast(xp: ModuleType):
     """Broadcast from input array with unknown shape"""
     x = xp.asarray([1, 2, 2])
diff --git a/tests/test_testing.py b/tests/test_testing.py
index 47eaa4d..10ce7ab 100644
--- a/tests/test_testing.py
+++ b/tests/test_testing.py
@@ -72,7 +72,8 @@ def test_assert_close_tolerance(xp: ModuleType):
 
 
 @param_assert_equal_close
-@pytest.mark.xfail_xp_backend(Backend.SPARSE, reason="index by sparse array")
+@pytest.mark.skip_xp_backend(Backend.SPARSE, reason="index by sparse array")
+@pytest.mark.skip_xp_backend(Backend.ARRAY_API_STRICTEST, reason="boolean indexing")
 def test_assert_close_equal_none_shape(xp: ModuleType, func: Callable[..., None]):  # type: ignore[explicit-any]
     """On Dask and other lazy backends, test that a shape with NaN's or None's
     can be compared to a real shape.
@@ -222,7 +223,7 @@ def test_lazy_xp_function_cython_ufuncs(xp: ModuleType, library: Backend):
     pytest.importorskip("scipy")
     assert erf is not None
     x = xp.asarray([6.0, 7.0])
-    if library in (Backend.ARRAY_API_STRICT, Backend.JAX):
+    if library.like(Backend.ARRAY_API_STRICT, Backend.JAX):
         # array-api-strict arrays are auto-converted to NumPy
         # which results in an assertion error for mismatched namespaces
         # eager JAX arrays are auto-converted to NumPy in eager JAX