Few years back I replaced the grip screws on my carry piece with some nice hex-drive ones(I don't know, because tactical or something). A few days back I'd removed the grips to clean under them and when replacing the screws the head neatly broke off the right bottom one.
Yes, I said words.
Far enough down couldn't get hold of it, so today got a 1/16" bit(new and sharp) and very carefully started center-drilling the screw, figuring either the bit would turn the screw or I'd have a hole to use a broken-screw extractor. Sure enough, when the bit got deep enough in the screw started turning, and was able to run it all the way through and out. Yay!
Now all I have to do is hope the original screws are still where I put them.
(I try to put any such parts in a baggie, labeled, and stick it in a particular place. I'm just hoping I didn't drop the damn things at some point while looking there for something else)