1.  Art Reblitz years ago suggested this excellent trick to me, which
I have used many times, for removing a screw broken off in wood:  File
saw-tooth points on the end of a brass tubing nipple slightly larger than
the screw.  Use this "hole saw" to drill a hole around the screw.  (The
screw often comes out with the plug off wood.  If not, it is easy to pull
out.)  Then plug and re-drill.

2.  If you can grab the screw or if one seems likely to break, first heat
it with a soldering iron to char the wood fibers around it.  This is
excellent for loosening frozen regulating (let-off) screws.  Too much
heating will loosen the hole, however.

3.  'Popular Mechanics' magazine of 1925 offers yet another trick: Drill
a hole adjacent to the screw.  Then, with a drift or punch, tap the screw
sideways into the new hole, which should be larger than the diameter of
the screw.  Now it is loose, and can be pulled out, whereupon the
adjacent hole is plugged.