Chris BeHanna advises:

1. Swear a few times.

2. Drill a hole.

3. Get the screw extractor.

4. You were too cheap to buy a tap wrench, so you turn the screw extractor with vice grips. This breaks the screw extractor.

5. Curse a blue streak.

6. Take the part to a machinist for EDM work.

1. Spray the part with Kroil before trying to get it out. This may save you from breaking it in the first place. Let it soak for a day.

2. Take a punch and, in the center of the screw head, rap sharply once with a hammer. This helps to break any cold welding that may have occurred.

3. Use an impact driver. Use the correct tip (hint: do not use phillips tips where Posi-Drive is required and vice versa). If the head gets chewed up, I've had some success cutting a slot for a common tip with a cold chisel and a hammer (but be careful not to mar what the screw is in!).

4. For really resistant pieces, gently heat the piece that the bolt is in. Be careful to not ruin the temper of the metal, if applicable (e.g. frames).