Noise and vibration occurring in one faucet only is an indication that the problem is right there in that specific faucet, and it is likely the washers. These are disc-shaped pieces usually made of rubber that fit inside the faucet assembly and help prevent leaks. These can wear out due to age or a bad washer seat, though Ace Hardware advises that the only real clue to whether the washer seat is bad is if the washers constantly have to be replaced.
Along with the possibility of certain parts of the faucet being bad comes the possibility that the entire faucet is bad or that something else in the set-up is loose. If the shut-off valve under the sink and the pipes leading to the sink faucet all seem to be secure, and replacing the washer didn’t work, it could be the faucet’s overall construction is simply shoddy and that replacing the faucet would stop the noise. While you don’t have to get a fancy, top-of-the-line faucet, you do need to look for one that has better construction. Look at consumer guides and ask friends which brands they use if they have not had a problem with their faucets.
If the vibrations are accompanied by sounds that are similar to banging, and they happen on many faucets or even every faucet, you could have a situation called water hammer or hydraulic shock. If you cut off the faucet when water is rushing toward it too quickly, the water can hit the closed valve and ricochet back and forth between the valve and the nearest bit of pipe wall that isn’t parallel to the water’s direction, causing noise and movement in the pipes. The remedy for this is installing something called a water hammer arrestor at each faucet, which provides extra space for the blocked water to head into, preventing it from flowing backwards.
Water rushing through a pipe can often start the pipe shaking because of the water’s force as it heads around corners and curves. It may look like the cut-off valve is involved because that’s obviously starting and stopping the flow of water through the pipe. If the pipe isn’t anchored correctly, it can move and cause vibrations, but that also means it can possibly hit anything near it. Locating the pipe and strapping it down can solve this, although you may have to call in a plumber to handle this as it would involve opening up the wall to find the pipe.