I have, in all honesty, never encountered any particularly overt sexism in the flying world. I know it exists; I've heard plenty of stories about Nimrod siggies and the suchlike, but I've not really seen any of it in the fast jet stream. I've never encountered any racism at all, to be honest. I've seen friendly banter directed at people for being Scottish, being female, being Muslim, being public school educated, having crap A levels, being overweight, having big ears, whatever; but never any of the genuinely malicious proper racism or sexism discussed so far. I know it's friendly banter because it was always amongst mates and never upset anyone.
Regarding sexism, I have seen a little bit of evidence that it exists in the Army. Chatting to Army mates about it, it seems to be built around the fact that the real "teeth" job in the Army; shooting people with guns; is for men only. The attitude persists that the pretty little girls are there to be stewards, AGC people, int officers, and the big strong men are there to do the "proper" job. That results in your impressionable 18 year old private thinking that he's a lot better than women because they can't do what he does. I've seen quite a nasty undercurrent of aggressive sexism in some Army people. I'm sure it's not necessarily fully institutionalised, but I've been disgusted at the behaviour and attitudes of some Army guys.
From my experience in the RAF, you can't sit there thinking you're better than women when you share a crewroom with a girl wearing a Harrier patch and a TLP patch... or if she's the STANEVAL pilot checking you, or the QWI doing your strafe check. There's the occasional bit of "lumpy jumper syndrome," but it's got a limit; while you might go through flying training with a girl who seemed to get a few extra breaks from the instructors, it can't be taken to the extreme. If she can't lead a four ship around, then she won't get her four ship lead tick, after all.