Removing screwed-up screws

When you can neither finish driving nor back out a screw, these two options save the day.

Stripped screw head: Burnish it and back it out

Using the burnishing tip of a Pro Grabit screw remover [Sources], bore a shallow, smooth countersink into the head to match the threaded extractor tip.
Backing it out

Flip the extractor end-for-end, insert the threaded tip in the burnished socket, and then use reverse gear, firm pressure, and slow speed to back the screw out of the wood. You can now drive a new screw in the same hole.
Broken screw: Bore around it, plug the hole, and start over

To use a hollow-tube extractor [B] in a handheld drill, first make a hardwood guide block by boring a hole matching the extractor's diameter.
Bore around screw

Center the guide block over the broken screw and clamp it in place. With the drill in forward gear, bore about ¹⁄₄ " deep. Remove the guide block.
Remove screw

Switch to reverse and bore until the plug either breaks free on its own, or you've reached a sufficient depth to pop it loose with a small screwdriver.
Plug hole

Glue in a dowel to fill the hole, leaving it slightly proud of the surface. Allow it to dry.
Drive a new screw

Sand the dowel smooth, redrill the pilot hole and countersink, and drive a new screw.

Sources
Pro Grabit screw remover: 3-piece set, no. 148984, Woodcraft, 800-225-1153, woodcraft.com.
Hollow-tube screw extractor: ¹⁄₄ " diameter, no. 124210, 516" diameter, no. 124211; 38" diameter, no. 124212, Woodcraft.