Our local simracing group has funraces on Thursday evenings and league events (or filler events between seasons) on Saturday evenings, this has been going on for a few years now, so those are the two I picked
Personally I try to only really play online races during the week if I can.
Weekends obviously fill up the servers with exponentially more people, which does make finding full lobbies of whatever you're looking for much easier, but in my experience that's usually a double edged sword. As more people tend to have the effect of you exponentially meeting more bad apples (especially when it's GT3's).
Just my opinion though, weekend racing can be great fun at times~