`singledispatchmethod.register` fails with `typing.Self` annotation

Bug report

Bug description:

When using singledispatchmethod, the register decorator function raises an exception if self is annotated with typing.Self:

TypeError: Invalid annotation for 'self'. typing.Self is not a class.

Minimal example:

from functools import singledispatchmethod
from typing import Self


class Foo:
    @singledispatchmethod
    def bar(self: Self, arg: int | str) -> int | str: ...

    @bar.register
    def _(self: Self, arg: int) -> int:
        return arg


foo = Foo()

print(foo.bar(1))

The workaround is to remove the typing.Self annotation for the _ function.

(Same for singledispatch but that should not be used in that case anyway)

CPython versions tested on:

3.13

Operating systems tested on:

Linux

Linked PRs

• gh-130829
• gh-143465

Start at functools.singledispatchmethod.register and reproduce the minimal example from the report on Python 3.13. Check the existing linked work before proceeding. Done means a method using typing.Self can be registered without the invalid-annotation TypeError, with regression coverage for the example.