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I was replacing the machine heads on Dad's old 12-string guitar (more on that later) - these are attached by 7 screws per side, most of which were badly rusted. As a result, when I tried to unscrew them, nearly all broke off below the surface of the wood, causing a bit of consternation.
The screws were hardened and only about 1/16" diameter, so there was no question of trying to drill them out. Cutting a slot wasn't an option either - not enough material, and I doubt they would've turned anyway. The commonest trick for removing stuff like this is to make a small tube drill (aka plug cutter, core drill etc.) to go over the broken screw. This removes a plug of material containing the old screw. The hole can then be plugged and a new screw hole drilled.
Various people have used this trick online - some have used standard plug cutters (there's a set of three, but I can't remember the link!) but most have made their own from tube. I made mine from a bit of steel rod (actually a nail!) with four hand-filed teeth. It doesn't have to be anything fancy. I could've made it from drill rod and hardened it, but I didn't feel bothered at the time. The outside diameter of the tube is about 3.7mm, from memory. I also made a little brass guide pin to locate the tube accurately over the old screw hole.
When drilling, stop frequently and remove the material that's built up inside the tube - I drill a hole right through so a pin can be shoved through to clean it. Also, watch you don't catch the old screw with the teeth. If you do, the teeth have a tendency to bend outwards, which rips up the wall of the hole when the drill is pulled out. I did this twice and had to make a new drill each time - however, it's pretty quick to do.
Here's some photos:
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 Some test holes |
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 Old screwholes in headstock |
 Drilled and plugged (light - beech, dark - teak) |
Quick video demo:
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