Many leaking faucets can only be repaired after you remove the handle, but unfortunately, age and rust can sometimes make this task impossible. When you have stripped a screw to the point that your screwdriver turns ineffectively in the screw slot, don't despair. Spray lubricant is often all you'll need to loosen stuck screws. For really stuck screws, you can drill a small hole in the head of the screw and use a drill bit to extract it. If that doesn't work, though, you may have to resort to a special bit called a screw extractor.
Spray a little thread lubricant on the head of a stripped screw and let the lubricant seep into the threads for about five minutes. Try unscrewing the screw again, using a screwdriver that is new, or at least is not worn.
Drill a small hole into the head of a screw you can't remove with a screwdriver. When the bit has sunk about 1/8 inch into the head, reverse the drill and push while you use the drill bit to unscrew the screw. If the bit slips, switch the drill to forward again and drill a little deeper, then switch it back to reverse and try again.
Use a screw extractor for screws that you cannot remove with a drill bit. Insert the head of the extractor into the hole created by the drill bit and switch the drill to reverse. Push on the screw as you pull the trigger of the drill and let the extractor unscrew the screw.
Use a larger drill bit to wear away the head of the screw if you cannot remove it with a screw extractor.
Rub a little abrasive hand soap onto the end of the drill bit to strengthen the grip.
Screw extractors are made of hardened steel that is very brittle, so don't over-torque them. If an extractor breaks off into the screw, you will have no recourse but to remove the head of the screw since the metal of the extractor is harder than any drill bit.