The phrase "on the weekends" can mean "most or many, not all", and is often shortened to "on weekends" or simply "weekends"
There is also the obvious phrase "this weekend", i.e. the next one coming or the one just past. "On the weekend" basically means exactly "this weekend". If another weekend farther off in time has been specified, "(on) that weekend" is used.
I believe "at the weekend(s)" is quite common in British English, but I don't know the subtleties.
Student or Learner