There are a couple tricks I use.
Torx tip, and a hammer, is one of the first, usually. Find the largest that will just fit in the stripped hole, and tap the driver tip in to the mangled socket with a small hammer. Leave it there, and carefully put the driver socket on to it, apply pressure to keep it from camming out. Usually works. Driving the tip in makes it fit the screw head very closely, and the shock often helps to break loose the locking compound, if there is any. Small hammer! 2-4 oz hammer head, tap tap tap!
Applying some grip compound can sometimes help considerably. They sell small tubes of the stuff but a can of valve grinding compound will last tree lifetimes.
Make nice with your Dentist and see if he or she has any old carbide dental burrs that are never going to be used. A Dremel Tool is nowhere near as fast turning as a dental handpiece, but you can use the burr to carve out the corners of the socket to fit a larger size driver tip. Or carve a slot, for a slot tip, or simply remove the head of the screw. They can be used to cut out pieces of broken taps too.
Buy quality screws to replace ALL the ones that are there, to avoid the repeat of the game.