Absolutely do not stand on the seat. Beyond getting dirty shoes all over the toilet, the asymmetic force during initiation is a great way to break the toilet seat or overcompress the wax seal beneath the toilet (I had an ex-roommate damage my toilet this way). Putting your feet on the rim is a good way to slip and fall. If you want to simulate a squat toilet with a Western fixture, raise up your feet on a step stool. Sit down first then raise your feet. </rant>
Sometimes you are in a public place with nothing to use as a platform. Never use the seat, of course, since other people put their butts there, and as you say they can be fragile. I use the rim. I have never slipped or fallen. It is quite secure.
I always joke with my friends when I tell them the girls in Asian countries always have more shapely asses than their Asian-American counterparts. That's because they work out their asses when they need to do their business.