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Continuing the work begun as part of issue #20182, this is regarding the Argument Clinic conversion of Modules/selectmodule.c.h.
I have a complete conversion ready and will create a PR momentarily.

See PR 4265.

While reviewing PR 4265, I noticed that the return type like "-> None" or "-> int" is removed from the function definition, so the PR removes this information. But it seems to be a limitation of Argument Clinic: I opened the issue #31939.
IHMO it's not a blocker issue. We can add it back later, once the issue #31939 is implemented.

When ready, should this (and other AC conversions) be backported to the relevant 3.x branches?

New changeset 6dc57e2a20c5beb99e8bf5eb04cc836d53fa9aee by Tal Einat in branch 'master':
bpo-31938: Convert selectmodule.c to Argument Clinic (GH-4265)
https://github.com/python/cpython/commit/6dc57e2a20c5beb99e8bf5eb04cc836d53fa9aee

New changeset 0e6e7a1e52a53090e33ebef13f8f1fbe9bec2375 by Tal Einat (Michael Osipov) in branch 'master':
bpo-34419: selectmodule.c does not compile on HP-UX due to bpo-31938 (GH-8796)
https://github.com/python/cpython/commit/0e6e7a1e52a53090e33ebef13f8f1fbe9bec2375

This appears to have broken pydoc of the select module, I've attached a patch
$ python3.8 -m pydoc select
Traceback (most recent call last):
 File "/usr/lib/python3.8/inspect.py", line 2004, in wrap_value
 value = eval(s, module_dict)
 File "<string>", line 1, in <module>
NameError: name 'EPOLLIN' is not defined
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
 File "/usr/lib/python3.8/inspect.py", line 2007, in wrap_value
 value = eval(s, sys_module_dict)
 File "<string>", line 1, in <module>
NameError: name 'EPOLLIN' is not defined
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
 File "/usr/lib/python3.8/runpy.py", line 193, in _run_module_as_main
 return _run_code(code, main_globals, None,
 File "/usr/lib/python3.8/runpy.py", line 86, in _run_code
 exec(code, run_globals)
 File "/usr/lib/python3.8/pydoc.py", line 2769, in <module>
 cli()
 File "/usr/lib/python3.8/pydoc.py", line 2731, in cli
 help.help(arg)
 File "/usr/lib/python3.8/pydoc.py", line 1971, in help
 elif request: doc(request, 'Help on %s:', output=self._output)
 File "/usr/lib/python3.8/pydoc.py", line 1694, in doc
 pager(render_doc(thing, title, forceload))
 File "/usr/lib/python3.8/pydoc.py", line 1687, in render_doc
 return title % desc + '\n\n' + renderer.document(object, name)
 File "/usr/lib/python3.8/pydoc.py", line 384, in document
 if inspect.ismodule(object): return self.docmodule(*args)
 File "/usr/lib/python3.8/pydoc.py", line 1183, in docmodule
 contents.append(self.document(value, key, name))
 File "/usr/lib/python3.8/pydoc.py", line 385, in document
 if inspect.isclass(object): return self.docclass(*args)
 File "/usr/lib/python3.8/pydoc.py", line 1352, in docclass
 attrs = spill("Methods %s:\n" % tag, attrs,
 File "/usr/lib/python3.8/pydoc.py", line 1298, in spill
 push(self.document(value,
 File "/usr/lib/python3.8/pydoc.py", line 386, in document
 if inspect.isroutine(object): return self.docroutine(*args)
 File "/usr/lib/python3.8/pydoc.py", line 1411, in docroutine
 signature = inspect.signature(object)
 File "/usr/lib/python3.8/inspect.py", line 3093, in signature
 return Signature.from_callable(obj, follow_wrapped=follow_wrapped)
 File "/usr/lib/python3.8/inspect.py", line 2842, in from_callable
 return _signature_from_callable(obj, sigcls=cls,
 File "/usr/lib/python3.8/inspect.py", line 2296, in _signature_from_callable
 return _signature_from_builtin(sigcls, obj,
 File "/usr/lib/python3.8/inspect.py", line 2109, in _signature_from_builtin
 return _signature_fromstr(cls, func, s, skip_bound_arg)
 File "/usr/lib/python3.8/inspect.py", line 2057, in _signature_fromstr
 p(name, default)
 File "/usr/lib/python3.8/inspect.py", line 2039, in p
 default_node = RewriteSymbolics().visit(default_node)
 File "/usr/lib/python3.8/ast.py", line 360, in visit
 return visitor(node)
 File "/usr/lib/python3.8/ast.py", line 445, in generic_visit
 new_node = self.visit(old_value)
 File "/usr/lib/python3.8/ast.py", line 360, in visit
 return visitor(node)
 File "/usr/lib/python3.8/ast.py", line 445, in generic_visit
 new_node = self.visit(old_value)
 File "/usr/lib/python3.8/ast.py", line 360, in visit
 return visitor(node)
 File "/usr/lib/python3.8/inspect.py", line 2031, in visit_Name
 return wrap_value(node.id)
 File "/usr/lib/python3.8/inspect.py", line 2009, in wrap_value
 raise RuntimeError()
RuntimeError

There is no need to change default-value signatures. The root problem is that select.epoll.register does not have the __module__ attribute.
Well, seems that all method objects do not have the __module__ attribute. There are two solutions:
1. Make inspect falling back to __objclass__.__module__.
2. Set __module__ for method objects.
But this is a separate issue.

New changeset d051801052211b533c46a593b1c1bccf649a171c by Anthony Sottile in branch 'master':
bpo-31938: Fix default-value signatures of several functions in the select module (GH-21066)
https://github.com/python/cpython/commit/d051801052211b533c46a593b1c1bccf649a171c

New changeset 60cbdc81d1bd5c175ff890ee4cfdc85c10090b75 by Anthony Sottile in branch '3.8':
[3.8] bpo-31938: Fix default-value signatures of several functions in the select module (GH-21066) (GH-21098)
https://github.com/python/cpython/commit/60cbdc81d1bd5c175ff890ee4cfdc85c10090b75

New changeset 48fc35ada7c4237f11bb9638387bfe5b4cebfb12 by Anthony Sottile in branch '3.9':
[3.9] bpo-31938: Fix default-value signatures of several functions in the select module (GH-21066) (GH-21097)
https://github.com/python/cpython/commit/48fc35ada7c4237f11bb9638387bfe5b4cebfb12

> There is no need to change default-value signatures. The root problem is that select.epoll.register does not have the __module__ attribute.
Oh, I saw your comment after merging Anthony's PR. At least, I confirm that it fix the reported issue: "./python -m pydoc select" doesn't fail anymore.
Maybe open a separated issue for the __module__ attribute issue.

pydoc doesn't show the eventmask parameter whereas it is in select.epoll.register.__text_signature__. I created bpo-41095: "inspect.signature() doesn't parse __text_signature__ containing a newline character".

Thanks Anthony Sottile for fixing pydoc select! The fix is now in 3.8, 3.9 and master branches.