diff --git a/pandas/conftest.py b/pandas/conftest.py
index d7e8fbeb9336b..14a107d563d62 100644
--- a/pandas/conftest.py
+++ b/pandas/conftest.py
@@ -619,6 +619,7 @@ def _create_mi_with_dt64tz_level():
     "tuples": MultiIndex.from_tuples(zip(["foo", "bar", "baz"], [1, 2, 3])),
     "mi-with-dt64tz-level": _create_mi_with_dt64tz_level(),
     "multi": _create_multiindex(),
+    "mixed-int-string": Index([0, "a", 1, "b", 2, "c"]),
     "repeats": Index([0, 0, 1, 1, 2, 2]),
     "nullable_int": Index(np.arange(100), dtype="Int64"),
     "nullable_uint": Index(np.arange(100), dtype="UInt16"),
diff --git a/pandas/tests/base/test_value_counts.py b/pandas/tests/base/test_value_counts.py
index 89f3c005c52f0..fc3743ffe150b 100644
--- a/pandas/tests/base/test_value_counts.py
+++ b/pandas/tests/base/test_value_counts.py
@@ -33,6 +33,7 @@ def test_value_counts(index_or_series_obj):
         with pytest.raises(NotImplementedError, match="float16 indexes are not "):
             expected.index.astype(obj.dtype)
         return
+
     if isinstance(expected.index, MultiIndex):
         expected.index.names = obj.names
     else:
@@ -47,21 +48,27 @@ def test_value_counts(index_or_series_obj):
 
     # TODO(GH#32514): Order of entries with the same count is inconsistent
     #  on CI (gh-32449)
-    if obj.duplicated().any():
-        result = result.sort_index()
-        expected = expected.sort_index()
-    tm.assert_series_equal(result, expected)
+    try:
+        if obj.duplicated().any():
+            result = result.sort_index()
+            expected = expected.sort_index()
+        tm.assert_series_equal(result, expected)
+    except TypeError:
+        pytest.xfail("dtypes not suitable for sorting.")
 
 
 @pytest.mark.parametrize("null_obj", [np.nan, None])
 def test_value_counts_null(null_obj, index_or_series_obj):
     orig = index_or_series_obj
     obj = orig.copy()
+    present_types = set([type(i) for i in obj])
 
     if not allow_na_ops(obj):
         pytest.skip("type doesn't allow for NA operations")
     elif len(obj) < 1:
         pytest.skip("Test doesn't make sense on empty data")
+    elif len(present_types) > 1:
+        pytest.skip("Test doesn't make sense on data with different index types.")
     elif isinstance(orig, MultiIndex):
         pytest.skip(f"MultiIndex can't hold '{null_obj}'")
 
diff --git a/pandas/tests/test_algos.py b/pandas/tests/test_algos.py
index 34a0bee877664..1fc326610d8d6 100644
--- a/pandas/tests/test_algos.py
+++ b/pandas/tests/test_algos.py
@@ -71,7 +71,10 @@ def test_factorize(self, index_or_series_obj, sort):
             expected_uniques = expected_uniques.astype(object)
 
         if sort:
-            expected_uniques = expected_uniques.sort_values()
+            try:
+                expected_uniques = expected_uniques.sort_values()
+            except TypeError:
+                pytest.xfail("dtypes not suitable for sorting.")
 
         # construct an integer ndarray so that
         # `expected_uniques.take(expected_codes)` is equal to `obj`