Where should the apostrophe go - mens' or men's?

"Men's" is the plural possive, as in the men's toilet (belong to men (plural).

MOST nouns in English take apostrophe-s for singular possessive, s for plural, and s-apostrophe for plural possessive.

Fred is a waiter. (singular)

This is the waiter's jacket. (singular possessive)

Fred and Sam are waiters. (plural)

These are the waiters' jackets. (plural possessive)

Because the plural of "man" is "men" without an s, the plural takes apostrophe-s.

Fred is a man.

This is the man's jacket.

Fred and Sam are men.

These are the men's jackets.