I take a proper towel because after a run I'm dripping with sweat and there is no way I'm prepared to go out in public without a shower and fresh underwear (if there is anything grosser than inner-thigh/butt crack sweat I don't know it) and if I'm doing a class I like a towel under me if I'm on the floor.

If they never break a sweat then a hand towel might be ok.