diff --git a/dask/dataframe/backends.py b/dask/dataframe/backends.py
index 9ef5d1b360f..5daaec7819b 100644
--- a/dask/dataframe/backends.py
+++ b/dask/dataframe/backends.py
@@ -703,8 +703,8 @@ def categorical_dtype_pandas(categories=None, ordered=False):
     return pd.api.types.CategoricalDtype(categories=categories, ordered=ordered)
 
 
-@tolist_dispatch.register((pd.Series, pd.Index, pd.Categorical))
-def tolist_pandas(obj):
+@tolist_dispatch.register((np.ndarray, pd.Series, pd.Index, pd.Categorical))
+def tolist_numpy_or_pandas(obj):
     return obj.tolist()
 
 
@@ -770,11 +770,13 @@ def to_backend(cls, data: _Frame, **kwargs):
 @make_meta_dispatch.register_lazy("cudf")
 @make_meta_obj.register_lazy("cudf")
 @percentile_lookup.register_lazy("cudf")
+@tolist_dispatch.register_lazy("cudf")
 def _register_cudf():
     import dask_cudf  # noqa: F401
 
 
 @meta_lib_from_array.register_lazy("cupy")
+@tolist_dispatch.register_lazy("cupy")
 def _register_cupy_to_cudf():
     # Handle cupy.ndarray -> cudf.DataFrame dispatching
     try:
@@ -786,5 +788,9 @@ def meta_lib_from_array_cupy(x):
             # cupy -> cudf
             return cudf
 
+        @tolist_dispatch.register(cupy.ndarray)
+        def tolist_cupy(x):
+            return x.tolist()
+
     except ImportError:
         pass
diff --git a/dask/dataframe/partitionquantiles.py b/dask/dataframe/partitionquantiles.py
index a0c24300dd2..0318d3d7f7a 100644
--- a/dask/dataframe/partitionquantiles.py
+++ b/dask/dataframe/partitionquantiles.py
@@ -79,6 +79,7 @@
 
 from dask.base import tokenize
 from dask.dataframe.core import Series
+from dask.dataframe.dispatch import tolist_dispatch
 from dask.utils import is_cupy_type, random_state_data
 
 
@@ -262,7 +263,7 @@ def percentiles_to_weights(qs, vals, length):
         return ()
     diff = np.ediff1d(qs, 0.0, 0.0)
     weights = 0.5 * length * (diff[1:] + diff[:-1])
-    return vals.tolist(), weights.tolist()
+    return tolist_dispatch(vals), weights.tolist()
 
 
 def merge_and_compress_summaries(vals_and_weights):
@@ -408,9 +409,6 @@ def percentiles_summary(df, num_old, num_new, upsample, state):
         Scale factor to increase the number of percentiles calculated in
         each partition.  Use to improve accuracy.
     """
-    from dask.array.dispatch import percentile_lookup as _percentile
-    from dask.array.utils import array_safe
-
     length = len(df)
     if length == 0:
         return ()
@@ -425,12 +423,22 @@ def percentiles_summary(df, num_old, num_new, upsample, state):
     elif is_datetime64_dtype(data.dtype) or is_integer_dtype(data.dtype):
         interpolation = "nearest"
 
-    # FIXME: pandas quantile doesn't work with some data types (e.g. strings).
-    # We fall back to an ndarray as a workaround.
+    # FIXME: Series.quantile doesn't work with some data types (e.g. strings).
+    # We try using DataFrame.quantile(..., method="table") as a workaround.
     try:
         vals = data.quantile(q=qs / 100, interpolation=interpolation).values
     except (TypeError, NotImplementedError):
-        vals, _ = _percentile(array_safe(data, like=data.values), qs, interpolation)
+        interpolation = "nearest"
+        vals = (
+            data.to_frame()
+            .quantile(
+                q=qs / 100,
+                interpolation=interpolation,
+                numeric_only=False,
+                method="table",
+            )
+            .iloc[:, 0]
+        )
 
     if (
         is_cupy_type(data)
diff --git a/dask/dataframe/shuffle.py b/dask/dataframe/shuffle.py
index 4698358495b..f0d00c4d3ed 100644
--- a/dask/dataframe/shuffle.py
+++ b/dask/dataframe/shuffle.py
@@ -120,13 +120,13 @@ def sort_values(
     df: DataFrame,
     by: str | list[str],
     npartitions: int | Literal["auto"] | None = None,
+    shuffle: str | None = None,
     ascending: bool | list[bool] = True,
     na_position: Literal["first"] | Literal["last"] = "last",
     upsample: float = 1.0,
     partition_size: float = 128e6,
     sort_function: Callable[[pd.DataFrame], pd.DataFrame] | None = None,
     sort_function_kwargs: Mapping[str, Any] | None = None,
-    **kwargs,
 ) -> DataFrame:
     """See DataFrame.sort_values for docstring"""
     if na_position not in ("first", "last"):
@@ -176,7 +176,7 @@ def sort_values(
                 f"Dask currently only supports a single boolean for ascending. You passed {str(ascending)}"
             )
 
-    divisions, mins, maxes, presorted = _calculate_divisions(
+    divisions, _, _, presorted = _calculate_divisions(
         df, sort_by_col, repartition, npartitions, upsample, partition_size, ascending
     )
 
@@ -193,6 +193,7 @@ def sort_values(
         df,
         by[0],
         divisions,
+        shuffle=shuffle,
         ascending=ascending,
         na_position=na_position,
         duplicates=False,
diff --git a/dask/dataframe/tests/test_shuffle.py b/dask/dataframe/tests/test_shuffle.py
index 80b2bc4311e..9df246f73e6 100644
--- a/dask/dataframe/tests/test_shuffle.py
+++ b/dask/dataframe/tests/test_shuffle.py
@@ -1482,6 +1482,22 @@ def test_sort_values(nelem, by, ascending):
     dd.assert_eq(got, expect, check_index=False, sort_results=False)
 
 
+@pytest.mark.parametrize(
+    "backend", ["pandas", pytest.param("cudf", marks=pytest.mark.gpu)]
+)
+@pytest.mark.parametrize("by", ["x", "z", ["x", "z"], ["z", "x"]])
+@pytest.mark.parametrize("ascending", [True, False])
+def test_sort_values_tasks_backend(backend, by, ascending):
+    pdf = pd.DataFrame(
+        {"x": range(10), "y": [1, 2, 3, 4, 5] * 2, "z": ["cat", "dog"] * 5}
+    )
+    ddf = dd.from_pandas(pdf, npartitions=10).to_backend(backend)
+
+    expect = pdf.sort_values(by=by, ascending=ascending)
+    got = dd.DataFrame.sort_values(ddf, by=by, ascending=ascending, shuffle="tasks")
+    dd.assert_eq(got, expect, sort_results=False)
+
+
 @pytest.mark.parametrize("ascending", [True, False, [False, True], [True, False]])
 @pytest.mark.parametrize("by", [["a", "b"], ["b", "a"]])
 @pytest.mark.parametrize("nelem", [10, 500])