Lowes (or maybe Home Depot) has a small "stripped screw remover". It's saved my sanity more than once.
One side is a tapered drill that profiles the hole in the head to a set angle...setting it up for the other end, which is a reverse drill bit. Jam it in and run your drill in reverse...it'll grab (not designed to remove material but rather to dig in and catch) and spin the screw out.
The only hard part is finding the small set...there's lots of larger ones.
Takes about 5 seconds from start to finish.
It will not grab a screw shank if you snap the head off...it needs enough material left in the head for the reverse drill teeth to catch and hold.
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