Before starting any work, you must shut off the water to the faucet. Look for two small valves, one for the hot water line and one for the cold, located in the cabinet directly beneath the sink. Turn the handles of both valves clockwise until they are fully closed. If you cannot find these local shutoff valves, you will need to turn off the main water supply valve for the entire house. Once the valves are closed, turn on both faucet handles to release any trapped water and pressure in the lines until the water stops flowing completely.

The faucet handle is typically attached with a single screw, which is hidden under a decorative cap on the top or front of the handle. Use a small flathead screwdriver or a utility knife to carefully pry this cap off, revealing the screw underneath. Use a Phillips head screwdriver to turn the screw counter-clockwise and remove it completely. Set the cap and screw aside in a safe place. Once the screw is out, you should be able to pull the handle straight up and off the valve stem. It may be stuck; if so, wiggle it gently from side to side to loosen it.

With the handle removed, you will see a hexagonal nut called the packing nut. This nut holds the valve stem in place. Using an adjustable wrench or channel-lock pliers, grip the packing nut and turn it counter-clockwise to loosen and remove it. Once the packing nut is off, you can remove the valve stem. Some stems can be pulled straight out, while others may need to be unscrewed from the faucet body by turning them counter-clockwise.

Inspect the valve stem you just removed. At the very bottom, there will be a rubber seat washer held on by a brass screw. This is the most common cause of a spout drip. Use a flathead screwdriver to remove this screw and the old washer. Replace it with a new seat washer of the exact same size. Also, examine the O-ring(s) on the body of the stem. If they appear flattened, cracked, or brittle, they should be replaced. Pry the old O-ring off with a small pick or screwdriver and roll the new one into its groove. Before reinserting the stem, apply a small amount of plumber's grease to the new O-ring(s) and the stem threads to ensure a smooth operation and good seal.

Before reinstalling the stem, look down into the empty faucet body to inspect the valve seat, which is the brass rim that the seat washer presses against. Mineral deposits or corrosion on this seat can cause a leak even with a new washer. Use a clean rag wrapped around a screwdriver to wipe the inside of the valve seat clean. If you feel nicks, pits, or roughness, you will need to smooth it. A specialized tool called a valve seat wrench or seat dresser is used for this. Insert the tool and give it a few gentle clockwise turns to lightly grind the seat down to a smooth, flat surface.

Reassemble the faucet by performing the disassembly steps in reverse order. Carefully insert the repaired valve stem back into the faucet body, turning it clockwise if it is the threaded type. Hand-thread the packing nut back on, then use the adjustable wrench to give it a final quarter-turn to snug it up. Do not overtighten. Place the faucet handle back onto the valve stem, aligning it correctly. Insert the retaining screw and tighten it with the Phillips head screwdriver. Finally, press the decorative cap back into place on the handle.

With the faucet fully reassembled, it's time to test your work. First, make sure the faucet handles are in the 'off' position. Slowly turn the water supply valves under the sink counter-clockwise to restore water flow. Turn them all the way to the open position. Now, closely inspect the faucet for any leaks around the handle base. Then, turn the repaired handle on and off several times, checking for smooth operation. Finally, turn the water off and watch the spout for several minutes to ensure the drip has stopped. If you see a small leak around the handle, use your wrench to tighten the packing nut by another 1/8 of a turn.