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http://www.python.org/dev/peps/pep-3151 define exception classes like ConnectionError and BrokenPipeError
We need to refactor stdlib to use that shiny new exceptions instead of checking errno values if possible.
Also maybe we need to add some absent exceptions.
I guess NotConnectedError for ENOTCONN, which should be ConnectionError subclass.

Right now I'm working on a patch. ;) A pack of issues were produced in process.

Please, feel free to add me in nosylist for any issue related to this one.

I would support adding ENOTCONN under the ConnectionError umbrella. It is caught for shutdown() calls in a few standard library modules. Here is a demo showing how to trigger it (at least on Linux):
from socket import create_connection, SHUT_RDWR
from socketserver import TCPServer, BaseRequestHandler
from threading import Thread
server = TCPServer(("", 0), BaseRequestHandler)
Thread(target=server.serve_forever).start()
client = create_connection(server.server_address)
server.shutdown() # Ensure server has closed client connection
server.server_close()
client.send(bytes(1))
>>> client.shutdown(SHUT_RDWR)
Traceback (most recent call last):
 File "<stdin>", line 1, in <module>
OSError: [Errno 107] Transport endpoint is not connected

Oh, I didn't know this issue. FYI I modified the subprocess module to use new specialized OSError exceptions: issue #23234 (changeset 0c5ae257966f).