The most challenging thing about removing the handle from a bathroom sink faucet is figuring out how it was installed in the first place. With all of the different looks available for a bathroom sink faucet there are only two ways that the handle is secured in place. In order to remove the handles, you will need to be able to identify which method is holding yours together, and then know how to take it apart properly.
Identify which kind of mount is being used to hold the handle on the faucet. If you have handles that are made of clear plastic with what looks like a lid on top, then you have the kind that screw down through the top. If you have handles that look like a lever then they are attached using a small Allen screw at the base of the handle.
Shut off the water supply to the bathroom sink. Look under the sink to see if the shut-off valves are at the base of the feed lines. If they are not there, then you will need to go into the basement and shut the water off to the house. When you are removing the handle from a bathroom faucet there can be water pressure built up. If you do not shut the water off, then parts could come flying off the handle and strike you.
Use a flat-head screwdriver to pop the top off of your handle if you have the plastic knobs. This applies to single and double faucets. The top will pop off easily so be careful not to damage it. If you have levers then examine the base to find the Allen screw that is holding it into place. Be sure to have your goggles on.
Unscrew the Phillips head screw holding your plastic knob in place, or use an Allen wrench to unscrew the Allen screw holding your lever faucet handle in place. The Allen screws vary in size, so have a complete set of Allen wrenches with you to find the proper one.
Lift the handle from its mount and then repeat the process for the other handle if you have a double faucet.