It would depend where in the world you live. 'Stay calm' and 'Remain calm' imply that the audience is already calm, but that there is a disturbance or alarm which is to be ignored. However, 'Keep calm' may bring back memories of a series of British movies by the 'Carry On Gang'.
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.