diff --git a/Lib/functools.py b/Lib/functools.py
index 77ec852805c104..305ceb450a71c4 100644
--- a/Lib/functools.py
+++ b/Lib/functools.py
@@ -740,6 +740,7 @@ def _compose_mro(cls, types):
     # Remove entries which are already present in the __mro__ or unrelated.
     def is_related(typ):
         return (typ not in bases and hasattr(typ, '__mro__')
+                                 and not isinstance(typ, GenericAlias)
                                  and issubclass(cls, typ))
     types = [n for n in types if is_related(n)]
     # Remove entries which are strict bases of other entries (they will end up
@@ -837,6 +838,9 @@ def dispatch(cls):
             dispatch_cache[cls] = impl
         return impl
 
+    def _is_valid_dispatch_type(cls):
+        return isinstance(cls, type) and not isinstance(cls, GenericAlias)
+
     def register(cls, func=None):
         """generic_func.register(cls, func) -> func
 
@@ -844,9 +848,15 @@ def register(cls, func=None):
 
         """
         nonlocal cache_token
-        if func is None:
-            if isinstance(cls, type):
+        if _is_valid_dispatch_type(cls):
+            if func is None:
                 return lambda f: register(cls, f)
+        else:
+            if func is not None:
+                raise TypeError(
+                    f"Invalid first argument to `register()`. "
+                    f"{cls!r} is not a class."
+                )
             ann = getattr(cls, '__annotations__', {})
             if not ann:
                 raise TypeError(
@@ -859,11 +869,12 @@ def register(cls, func=None):
             # only import typing if annotation parsing is necessary
             from typing import get_type_hints
             argname, cls = next(iter(get_type_hints(func).items()))
-            if not isinstance(cls, type):
+            if not _is_valid_dispatch_type(cls):
                 raise TypeError(
                     f"Invalid annotation for {argname!r}. "
                     f"{cls!r} is not a class."
                 )
+
         registry[cls] = func
         if cache_token is None and hasattr(cls, '__abstractmethods__'):
             cache_token = get_cache_token()
diff --git a/Lib/test/test_functools.py b/Lib/test/test_functools.py
index 7bc355ff7213ee..9343d4e9fa3168 100644
--- a/Lib/test/test_functools.py
+++ b/Lib/test/test_functools.py
@@ -2665,6 +2665,74 @@ def f(*args):
         with self.assertRaisesRegex(TypeError, msg):
             f()
 
+    def test_register_genericalias(self):
+        @functools.singledispatch
+        def f(arg):
+            return "default"
+
+        with self.assertRaisesRegex(TypeError, "Invalid first argument to "):
+            f.register(list[int], lambda arg: "types.GenericAlias")
+        with self.assertRaisesRegex(TypeError, "Invalid first argument to "):
+            f.register(typing.List[int], lambda arg: "typing.GenericAlias")
+        with self.assertRaisesRegex(TypeError, "Invalid first argument to "):
+            f.register(list[int] | str, lambda arg: "types.UnionTypes(types.GenericAlias)")
+        with self.assertRaisesRegex(TypeError, "Invalid first argument to "):
+            f.register(typing.List[float] | bytes, lambda arg: "typing.Union[typing.GenericAlias]")
+        with self.assertRaisesRegex(TypeError, "Invalid first argument to "):
+            f.register(typing.Any, lambda arg: "typing.Any")
+
+        self.assertEqual(f([1]), "default")
+        self.assertEqual(f([1.0]), "default")
+        self.assertEqual(f(""), "default")
+        self.assertEqual(f(b""), "default")
+
+    def test_register_genericalias_decorator(self):
+        @functools.singledispatch
+        def f(arg):
+            return "default"
+
+        with self.assertRaisesRegex(TypeError, "Invalid first argument to "):
+            f.register(list[int])
+        with self.assertRaisesRegex(TypeError, "Invalid first argument to "):
+            f.register(typing.List[int])
+        with self.assertRaisesRegex(TypeError, "Invalid first argument to "):
+            f.register(list[int] | str)
+        with self.assertRaisesRegex(TypeError, "Invalid first argument to "):
+            f.register(typing.List[int] | str)
+        with self.assertRaisesRegex(TypeError, "Invalid first argument to "):
+            f.register(typing.Any)
+
+    def test_register_genericalias_annotation(self):
+        @functools.singledispatch
+        def f(arg):
+            return "default"
+
+        with self.assertRaisesRegex(TypeError, "Invalid annotation for 'arg'"):
+            @f.register
+            def _(arg: list[int]):
+                return "types.GenericAlias"
+        with self.assertRaisesRegex(TypeError, "Invalid annotation for 'arg'"):
+            @f.register
+            def _(arg: typing.List[float]):
+                return "typing.GenericAlias"
+        with self.assertRaisesRegex(TypeError, "Invalid annotation for 'arg'"):
+            @f.register
+            def _(arg: list[int] | str):
+                return "types.UnionType(types.GenericAlias)"
+        with self.assertRaisesRegex(TypeError, "Invalid annotation for 'arg'"):
+            @f.register
+            def _(arg: typing.List[float] | bytes):
+                return "typing.Union[typing.GenericAlias]"
+        with self.assertRaisesRegex(TypeError, "Invalid annotation for 'arg'"):
+            @f.register
+            def _(arg: typing.Any):
+                return "typing.Any"
+
+        self.assertEqual(f([1]), "default")
+        self.assertEqual(f([1.0]), "default")
+        self.assertEqual(f(""), "default")
+        self.assertEqual(f(b""), "default")
+
 
 class CachedCostItem:
     _cost = 1
diff --git a/Misc/NEWS.d/next/Library/2021-12-11-15-45-07.bpo-46032.HmciLT.rst b/Misc/NEWS.d/next/Library/2021-12-11-15-45-07.bpo-46032.HmciLT.rst
new file mode 100644
index 00000000000000..97a553d7ba29fb
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2021-12-11-15-45-07.bpo-46032.HmciLT.rst
@@ -0,0 +1,5 @@
+The ``registry()`` method of :func:`functools.singledispatch` functions
+checks now the first argument or the first parameter annotation and raises a
+TypeError if it is not supported. Previously unsupported "types" were
+ignored (e.g. ``typing.List[int]``) or caused an error at calling time (e.g.
+``list[int]``).