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I have on several occasions broken screw heads in pieces where I can't readily access or cut into the joint they're securing. Is there such a thing as a tiny core drill bit for cutting around what's left of the fastener so I can remove the rest of the wood piece it was securing?
Or is there any other technique for dealing with this situation when I can't cut the wood piece loose from behind using a reciprocating saw and can't just tear the piece free?
Yes, there are hollow screw extractors like you describe, such as these two examples:
Unfortunately, they may be difficult to get started if you're unable to fit the part into a drill press and you aren't able to make a pilot hole the size of the extractor.
In that case, you can try making a guide of the appropriate diameter and clamping it to your workpiece, or try the more typical type of screw extractor that Matt mentioned.
Sounds to me like you are describing a use for screw extractors
(source: asklaptopfreak.com)
Your drill needs to have the torque in order to get them in what's left of the head or shaft. They are drilled forward the reverse direction (lefty loosy) so that when the bit gets enough catch on the screw it gets pulled out.
It can be harder to use these when you don't have a head to work with but they could still work if you can't find what Rob suggested (Which is a good answer).
I think the best method I've seen to remove small screws with sheared-off heads is to drill access holes on opposite sides of the shaft, just large enough to accept the tips of a needle-nose pliers, get a firm grip on the shaft and twist it out. Obviously this marrs the wood noticeably but if you use pliers with very fine tips (if necessary by grinding them even smaller) the damage isn't too bad.
With brass in particular it is possible to drill right through the screw itself with a small bit, basically turning what remains into shavings. It's difficult to do by hand, I've only ever done it successfully once, but with a drill press this is much more viable (assuming the workpiece can fit the bed of the drill of course). So possibly kept as the technique of last resort.
