Well, I’ll start by saying that where I board we have an indoor arena. So, the outside temps may be colder than in a closed up arena (where the “feels like” temperature isn’t as much of a factor). There’s about 15" of snow outside and it’s just too deep for riding (for about the last 6 weeks now).
Before this winter, my cut-off used to be around 12 degrees F. This year, we got so accustomed to the cold that a few times I left the barn after riding, got in my car, and saw that the temps were 4 (or something similar). So I guess my cut-off standard has been lowered this year (or my brain was partially frozen).
However, when I do ride, I spend a lot of time walking before I pick up the trot. On really cold days, I don’t canter much. I have read that a slow warm up is important for your horse in cold weather. But I do admit that it’s harder for the rider to stay warm at the walk! I do use a quarter sheet on my horses–and I sometimes use 2 (and take one off after my warm up). Neither of my horses are clipped–but they are blanketed and don’t have much coat.
There are a few boarders that don’t ride at all in the winter, but most of us do. Some of the riders don’t ride when it’s under 15 F. Most of the lessons will be canceled if it’s below below 10 F (whereas last winter they would get canceled when it was below 15). 
