using a screw extractor
Time needed: 10 minutes
A screw extractor is actually two different parts working together: a screw extractor bit and a tap wrench. Screw extractor bits look like normal drill bits, except that they're roughly pyramid-shaped -- the tips are pointy, and they flare wider towards the base. The base has a square protuberance, which fits into the tap wrench. The interesting part about the screw extractor bit is that the threads on them run "backwards", or counterclockwise (drill bit threads run clockwise). You'll you use a normal drill bit to drill a hole in the bolt, and the push the corresponding screw extractor bit in -- the counterclockwise action used to screw the extractor bit into the hole will be the same counterclockwise action normally used to remove a bolt.
Here's a picture of the drill/extractor bits kit that I have. It includes five drill bits, and the corresponding five screw extractors. The drill bits are on the left side of the case; the screw extractors on the right.
Every report I've read about screw extractors says to use a tap wrench with the extractor bit, and not a regular box wrench. Apparently the extractor bits are easily broken when using a normal wrench -- I don't know for sure, because I didn't risk it, but I pass on the warning. The tap wrench worked perfectly for me; I bought one with a T-handle, which adjusts to hold very small bits all the way up to 1/2" bits (you'll want to compare the measurements on the extractor box with those on the tap wrench box). The end looks sort of like a drill chuck; you just slide your extractor bit in and turn a little collar to tighten it. That's all.
Now you're ready to drill. Put on your happy eye protection, attach the correct size drill bit, line it up smack dab in the middle of that stuck bolt, and drill. You don't need to drill all the way through the bolt; halfway through or less should suffice. Remember that you don't want to damage any non-bolt part of the bike, so it's much better to drill too little at first and then go back and drill more than it is to be over-enthusiastic and bust something. If you're in a metal shavings sensitive area, you might find it helpful to have a friend hold the tube end of a vacuum cleaner nearby to suck away stray fillings while you're drilling.
Once your hole is drilled, attach the corresponding screw extractor bit into your tap wrench. Simply insert the tip of the extractor bit into the hole that you just drilled, and turn counterclockwise. Since the threads are reversed on the extractor bit, turning it counterclockwise will tighten the bit into the hole in the bolt. The tapering on the bit means that it will eventually fit snugly within the broken bolt, at which point, turning the bit counterclockwise will force the bolt to turn counterclockwise as well. As we know with bolts, it's "righty tighty, lefty loosey", and turning a bolt counterclockwise will remove it. Huzzah! The bolt comes out, and you're happy. Now you can go reorder that stupid bike-specific bolt for ten dollars from the local dealer. Yay!