Looking on the internet deeply has found these results:

on the weekends is the most popular phrase on the web.

at weekends 13,790,000 results on the web	on the weekends 116,300,000 results on the web More popular!
Some examples and use cases from the internet:	Some examples and use cases from the internet:
It doesn't look like this phrase is very popular! The search could not find any examples on the internet.	Worked two jobs, National Guard on the weekends. I could work for Vince during the week and then write on the weekends. You know, something just on the weekends. He studies guitar and composition, and he deejays in Chinatown on the weekends. Spends school nights down there and comes home on the weekends. That's' cause they say that it rests on the weekends. I became a superdad on the weekends with Scott. Hitler takes countries on the weekends.

at weekends

13,790,000 results on the web

on the weekends

116,300,000 results on the web

More popular!

It doesn't look like this phrase is very popular! The search could not find any examples on the internet.

Worked two jobs, National Guard on the weekends.
I could work for Vince during the week and then write on the weekends.
You know, something just on the weekends.
He studies guitar and composition, and he deejays in Chinatown on the weekends.
Spends school nights down there and comes home on the weekends.
That's' cause they say that it rests on the weekends.
I became a superdad on the weekends with Scott.
Hitler takes countries on the weekends.