One of my bathroom sink faucets is leaking. It started leaking from just the cold water, then the hot water started leaking too. The leaks are getting progressively worse.
I want to take the handles off to check the cartridges and washers, but can't figure out how to get them off. I have tried putting a small screwdriver into the hold and turning and just pulling up on the handles but have had no luck.
It's up to you whether you want to change out the whole faucet or not. If you like that style, or don't have the ready cash, you might as well try to repair it.
The first step is to shut-off the water supply to the faucet under the sink. Then open the handles to release an other water under pressure in the pipes. Now put the stopper or a rag in the drain hole. You don't want to lose anything down there. Now it is time to remove the handles! It'll be pretty easy to do if you have the right sized Allen key to remove the set screw. Looks like a 3/32 or 5/64 to me. Pull the cartridge out (I'm not that familiar with the faucet in your photo, so there might be a little more
work & tools involved). Repeat with the other side.
Then bring them down to the local big box and pick-up your replacement cartridges and reverse the above instructions.