Neither. At the weekend. At weekends. Or even better, no preposition. What do you do weekends? What are you doing this weekend?

at weekend : British on weekend: American Both Correct

They are both wrong. You need "the" if it's singular. UK English: at the weekend US English: on the weekend

Australians say: On the weekend I am going to the beach/football etc

moorche's answer is correct. Occassionally, you might see "at" used this way: "At week's end, he hadn't even begun the assigment" - meaning "by the weekend". It refers to the start, or arrival of the weekend, rather than the whole weekend itslef.