Squealing taps (when turned off)

My bathroom faucets (hot/cold) often make a high-pitched squealing noise. The noise starts sometimes when the faucet is turned off, or the shower is turned off, or other faucets in the flat are turned off. The noise can be stopped by turning on the taps and turning them off again, or by leaving the water running slightly. Is this a simple washer replacement issue, or is something else going on? It seems strange that the noise only happens when the taps are turned off, rather than when the water is running.