In the UK, we say "at the weekend"; "on the weekend" sounds strange. I believe that Americans find "at the weekend" strange!

Forget the "on weekend." That's out. That's not English. The other two are fine. "On the weekend" and "on weekends" are pretty close, and they could be synonymous depending on the context "I take the class on the weekend" or "on weekends" is fine either way. The only difference is that "on the weekend" could refer to only one weekend only, not more than one, where "on weekends" refers always to more than one. The context would tell you if you're talking about a one time only weekend event or something occurring on 2 or more weekends.

Also, "at the weekend" is a good one to know. A: What did you do at the weekend? B: Hm, not much much really.

It is clear now. Thank you!

On weekend is incorrect. On the weekend, I ... Specifies during the coming weekend or a specific weekend if followed by a date. Ex. On the weekend of the 25th, I will be out of town. On weekends, I ... Generalizes the statement. All weekends.