I never have had any luck with the various screw extractors on the market. I most recently tried a Moody tool from this set. (It was a holiday gift) It did not work.
Occasionally, I have been able to drill out the center of a stuck screw, and poke loose the remaining part with a probe.
Some times the drill will get stuck to the screw. I stop the drill immediately, and hand turn it backwards, and often the screw comes out with it.
Several folks have mentioned left hand bits. Sounds like they would be worth a try.
In my case, The screw heads are usually broken off because the threaded body was stuck in it's hole, and someone (usually me) continued to turn it. If the body is stuck enough to break off the head, I guess it would be kind of tough for a tool to remove the body.