I thought the point of this thread was a general reference list of all the types of building that COULD be found in a settlement. It isn't really relevant whether ordinary schools existed or not; some towns MIGHT have had schools, and so they should be included in the list.

As an aside, I also strongly suspect that schools did exist in towns in the past, they just weren't as common as we know them today. I haven't verified this, so it might be wrong, but: in Muslim society, children would often go to the madrasa, where they would receive instruction in both religious and non-religious matters; schools have existed in Christian countries for centuries, and I would be very surprised if the same wasn't true of China, India and Japan, not to mention other places.

One thing to consider, though, is that schools were not always located in schoolhouses. I could easily imagine that in some rural village in, say, England, a charitable vicar would volunteer to teach the children inside the Church or Town Hall. It was a less formal, more fluid arrangement than now.