Here's a 30 second trick for removing a broken screw without damaging the surrounding wood!
Step 1: Frustration
Begin by noticing that your screw is suddenly spinning freely. After a moment of confusion comes the sinking realization that the screw has broken. Often this happens as you're driving when the screw is almost flush with the wood. If enough of the screw head is exposed you can carefully grip the screw with a small pair of pliers and remove the broken piece by gently twisting until the top of the screw can be lifted free.
If a screw breaks when the screw head is almost flush with the surface, take a moment to just breathe and feel frustrated.
Broken screws often happen in the finishing steps of a project (when you're working with slender delicate finishing screws or screws that came with a hinge or hardware set). It's tempting to give into your frustration and try any number of brute force methods- I've seen experienced woodworkers attack broken screws with everything from chisels to hammers in the heat of the moment. While you might dig the screw out, you'll likely damage the surrounding wood and have to spend even more time repairing the cosmetic damage.
Step 2: Apply Superglue
Find a spare bit that fits securely into the broken screw. If the screw was a softer metal it may have partially or fully stripped before breaking. Try a few bits to find one that makes the most contact with the geometry of the screw (try a small square bit in a badly stripped phillips screw).
Put a drop (just a drop!) of superglue into the screw and press the spare bit into the glue.
Let the glue cure for about 30 seconds- you just need the glue to tack, you don't want to leave it for 24 hours and let it fully cure.
Step 3: Gently Twist and Lift the Bit
GENTLY twist and wiggle the bit with your fingers while applying upward pressure. You're fighting the friction between the shaft of the screw and the wood so smaller, gentle movements will be more effective than sharp jerks.
Step 4: Reclaim the Bit
Once you're removed the broken screw, tap the broken screw against a the edge of a workbench or other hard surface. Since the screw is most likely much cleaner than your bit, the superglue should remain in the screw and your bit will be ready for use again! If some glue remains on the bit you can knock it off with a blade or a small flat head screw driver.
Step 5: Evaluate Your Next Steps
Be Open to Creative Adaptations-
You're removed the broken head of the screw but you still have a visible hole and the shaft of the screw imbedded in your project. It is very difficult to remove the rest of the screw without damaging the surrounding wood. It sounds obvious but remember that there is still most of a screw in the bottom of the hole- don't try to drive another screw into the same hole!
If the hole is in a highly visible place, you can fill the hole with wood putty or adjust the rest of your project to hide the hole or lessen its impact- perhaps you can shift a set of hinges slightly to cover it or switch to a larger trim board.
Evaluate the Probability of More Broken Screws-
If you broke the first screw in a set easily or if screws start breaking frequently, try lowering the torque on your drill, using a screw driver instead of a power drill, drilling a larger pilot hole, or switch to different screws (try shorter screws if your geometry allows or use equivalent screws made from a stronger material). It can be difficult (especially when the end of a project is in sight) to take the time to order different screws or find alternative tools but it might save you a lot of frustration and more broken screws.