Things You'll Need
Phillips #2 screwdriver
Drill w/ assorted small bits
Corroded or stripped screws can be a problem until you know the secret to removing them. Even if the screw is completely broken off, you can still remove the old shaft using a device known as an extractor. So take heart in the fact that you do not have to call a plumber to repair and replace old handles, and you don't need to hire a contractor to remove corroded or rusty screws, or screws where the head has been stripped out, but can do these tasks yourself in only a few minutes.
Manual Screw Removal
Use the screwdriver to try removing the screw first. If that does not work, your next step is to try removing the screw manually. To do this, pick out a drill bit that is approximately the same size as the screw shank. Use the drill to cut into the center of the screw head. Once you have drilled past the head, it should fall off. If it does not fall off, use the next larger size of drill bit and try again. Alternately, use pliers or vicegrips to twist the head off after drilling through it.
Pull the handle straight out and off of the screw shank. If the handle is corroded onto the faucet mechanism, tap it a few times with the vicegrips to loosen it up. If that still does not work, place the vicegrips, handle-first, behind the faucet handle and use them for leverage to pry the handle forward.
Adjust the vicegrips so that they can be clamped onto the protruding screw shank. Clamp them onto shank so that the adjustable screw is facing away from the direction of removal. In most situations, faucet screws are removed by turning them counter-clockwise. Apply pressure to the screw shank and it should release and unscrew from the faucet. If the screw is badly corroded, try tapping the vicegrips with another object while keep steady pressure on them, pushing counter-clockwise.
Use a Screw Extractor
Remove the screw head and faucet handle as described in Steps 1 and 2.
If the screw shank is too short to grip with vicegrips, you can use a screw extractor. Screw extractors resemble drill bits, but the spiraled shaft is grooved in a counter-clockwise direction, and the grooves become more widely spaced higher along the extractor shaft.
Select a drill bit that is smaller than the diameter of the screw shank, and drill 1/8-inch into the shank of the screw. Select an extractor that will barely fit into the hole you have drilled.
Screw the extractor into the hole in a counter-clockwise manner until it grips the inside of the screw shank.
Use vicegrips or a low-speed, low-torque drill and slowly turn the extractor. Do not use a high-speed drill as it will not usually work and may damage or break the extractor. As it tries to bore into the screw shank, the extractor will grip the screw tighter and the screw can be backed out of the hole.
Always wear safety glasses when drilling metal. The tiny slivers of metal are extremely dangerous, and can be blown into your eyes by an unexpected gust of breeze.