A faucet screw stripped usually means the handles are porcelain and the screw is recessed under the cap. Assuming everything else has been tried, it’s easiest to drill off the head. It’s the quickest, surest way of saving the handles.
Start with a small drill bit, about 1/8 of an inch. Only drill 1/8″ deep and stay as close to center as possible. Run the drill at high speed and be careful not to push very hard. Some finesse is required, or the bit can grab in the screw head and break.
Once the pilot hole is drilled, chose a drill bit close to the diameter of the screw. A #10 screw would need a 3/16 drill bit. Drill just below the head of the screw and the head should spin off. Sometimes a larger drill bit is needed, but do not drill too deep.
Once the head is gone, remove the handle. Stubborn handles will require gentle tapping and vinegar or lime remover to extract. Sometimes we use boiling hot water to help loosen the handle. When all else fails, get a faucet handle puller to remove stubborn handles.
Once the handle is removed, try to remove the remaining shaft of the screw with a vise grip pliers. If it comes out, you are in luck. Replace it with a brass or stainless screw so this does not happen again.
Quite often faucet manufactures use steel screws instead of brass ones and they rust in place. They are the ones that crumble (disintegrate) when we grab them with the vise grips.
When that happens, the remnants of the screw will have to be removed from the faucet stem. Two basic methods are to use an easy out, or to drill-and-tap the remaining screw.