Been a little enthusiastic with your screwdriver or has the bolt gotten corrosion welded into the hole? Snapped that delicate bolt off in the hole leaving you with a piece of £200 junk? Having been faced with this dilemma myself I investigated several methods for getting the screw out.
The screw was a 6mm aluminium bolt which had snapped about 2mm below the level of the thread-hole after too much torque.
1. Hit it with a hammer - Well actually a dot punch.
Trying to gently spin the bolt with a hammer and the punch didn't work and only put me at the peril of slipping with the punch and destroying the threads in the hole. But with the right break, for example one with a jagged surface, it might work.
2. Superglue!
Next thing was to try and superglue a nail to the top of the break and use that to spin out the bolt. In retrospect this was a very stupid move as it run the risk of having the very fluid glue run down into the threads and forever bind the bolt and hub in matrimony. Superglue isnt a filler and really only works when the two surfaces are smooth. It also isnt resistant to shear forces which are the kind of forces you will be appling to get the bolt out. This method might work with a very small drop of a filling glue like epoxy resin on the head of the nail.
3. Power-tools
Time to break out the big boys, so I wandered down to B&Q Wandsworth and picked myself up a high speed drill and some 1.5mm drill bits. For me the cost of the drill was far outweighed by the cost of the thing I was trying to salvage but obviously you could borrow a drill or a Dremel off someone to get this done. The drill was a budget single speed 700 RPM unit so i had to be very careful when starting the hole not to skid and kill the threads. I drilled two holes about 1.5mm into the top of the broken bolt. Although it wasnt planned the holes were close enough to touch in the centre leaving me with a sort of slot into which i put a small blade screwdriver and screwed the remains of the bolt out.
That ended the adventure for me but some other methods I tracked down which could be worth the effort if the equipment is valuable enough
A pencil eraser - Use the eraser on the end of a pencil to spin the bolt out. It could work if the bolt was loose enough.
A Screw extractor kit - This is an extension of the power-tools angle and involves drilling a single hole into the top of the broken bolt and then putting a device called a screw extractor in. The extractor is a tapered hardened steel bolt with a reverse thread on it. As you screw the appropriate sized extractor in, it gets tighter in the direction that you want to spin to get the broken bolt out. Most extractors seem to have flanges at the top to allow you to attack the problem with a little brute force and a spanner once the extractor is in place.
This method might require a workbench and a vice to hold the problem in place while you deal with it.
Differential heating( AKA) Flames - For parts which are subject to heating within their normal lifespan, for example car engine bolts you can use a gas blowtorch to apply heat to the surrounding area and maybe unfreeze that stubborn bolt by getting a bit of expansion in the surrounding threads.