"on most weekends" is correct and usable in written English.
You can use it to describe a recurring action or event that happens on the majority of weekends. For example, "I usually go out with my friends on most weekends."
On most weekends, odds are good that they'll be hosting at least one solid live concert in the back room, which has its own bar for ordering tallboys like they're going outta style.
Decibel level can be quite high when the restaurant fills, which happens on most weekends.
On most weekends, they hear music blasting from the homes they are sandwiched between.
-- and on most weekends they travel to other competitions, often out of state.
Freeride sessions are for children 12 to 16 on most weekends and holiday periods.
There's something going on most weekends, like team roping and barrel racing.
There's live music on most weekend nights, plus jazz on Sundays.
One hundred dollars wouldn't even cover a Broadway spectacle, but on most weekend nights, the city's theater scene offers plenty of riches for $20 or less.
As on most weekend afternoons since last November, they gathered on an unseasonably warm Saturday two weeks ago for a day of sailboat racing in the forefront of the Manhattan skyline.
On most weekend afternoons in downtown Summit, just to the west, shoppers can barely find a seat at Starbucks; drivers have to thread their way through the upscale shopping district a few times before finding a parking space.
The place is the Mid-City Lanes Rock 'n' Bowl, and on most weekend nights the sounds of scattering pins and knee-slapping zydeco reach to the parking lot.
