Are you imaginining standing on the seat, standing on the rim, or standing on the ground in a martial arts horse stance? Because I can see significant possible problems with all of these. (technically, problems with the last would be due to "lack of muscle conditioning," but not many people are conditioned well enough to stay in a horse stance for long).

The rim. The seat, provided it is solidly built and attached. Anecdotally I feel quite stable either way.

Edit: the trick to perching on various narrow surfaces, for me, is to have the heel on one side an the ball of the foot on the other. "Gripping" as it were.