A drip almost always means a worn cartridge, washer or O-ring. The trick is getting the exact match.
In two-handle faucets, drips come from a worn washer, O-ring or cartridge inside the hot or cold side. Difficulty: Level 3: Confident DIYer. Plan on 30 to 60 min.
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Turn off both shutoff valves under the sink and open the faucet to release pressure. Plug the drain so small parts can't fall in.
To find the leaking side, close only the hot valve. If the drip stops, it's the hot side; otherwise it's the cold.
Pop off the handle cap, remove the screw and the handle, then unscrew the stem or cartridge with a wrench.
Take the old part to the store or match the faucet brand and model. Replace the washer, O-ring or whole cartridge.
Coat the O-rings with plumber's silicone grease, reassemble, and turn the water on slowly.
Single-lever faucets drip when the cartridge inside wears out or its seals harden with age. Difficulty: Level 3: Confident DIYer. Plan on 30 to 60 min.
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Turn off both shutoff valves and open the faucet. Plug the drain.
Remove the handle. The set screw is usually under a small cap and takes a hex key.
Remove the retaining nut or clip and pull the cartridge straight out. Take a photo of its position first.
Buy the exact replacement for your faucet brand and model.
Install it in the same position, reassemble, and test.
You can't safely open a faucet without working shutoff valves. Turn off the main water if you need to, and have a plumber replace the valves.
Heavily corroded faucets can crack when forced. At that point replacing the whole faucet is often the better fix, either as a bigger DIY job or with a plumber.