Always plural, never singular, unless you're referring to a particular woman, man, lady, etc.
Gents' (plural, then apostrophe!) is BE only, and dated or humorous as far as I know.
Gentlemen's is very traditional and old-fashioned and is seldom used any more; you might see it for a very traditional BE shop, like a tailor.
Ladies' is also traditional and old-fashioned, but is still used by many people as if it were the corollary to men's. It isn't, because it really means 'nice,' socially respectable women, not all women; unlike men's, it implies a value judgment based on class or behavior.
Women's is more modern and inclusive, because it's really a neutral description, the equal counterpart to men's. It's like saying Frauen- instead of Damen-.
However, in AE at least, special terms are still used in the clothing industry which are anything but modern and inclusive.
juniors = girls in their teens and twenties
petites = short women
misses = standard-sized women
women = large, heavy women
So if you ask for women's clothing or women's shoes, that's fine, it would be the opposite of men's clothing and men's shoes.
But if you ask for women's sizes, you'll probably be sent to the large-size department.
PS: What's a.s.o. supposed to mean? I don't think it's English.