It is pretty common in the U.S. The number of swats has decreased in recent years, however. In my day, in HS the minimum by a principal was five. Now it seems to be two or three. Teachers, in my day, could deliver three swats. Now I think most places don’t allow teachers to deliver swats.
In public schools, the paddle is almost always used. Private schools vary a little. I’ve never heard of a cane being used. The paddles are usually quite heavy, and I think are designed to penetrate several layers of denim, the most common material worn.
Swats are ALWAYS with clothes ON. I’ve never heard of it any other way. I can’t even remember the threat or rumor of it bare, at school.