quarter sheets are good (as long as your horse doesn't think it's a dragon if the wind blows!). barefoot horses do best imo for riding in snow. make sure it's snow and not ice - hard packed snow can be okay as long as we don't have those temp swings where it goes from frost to thaw and back and forth because that will make even the hard packed snow slippery.
i've ridden w/t/c in snow, and even jumped up to 3' in snow. but that was on level ground, the temps had stayed consistently well below freezing since the snowfall, and while it packed down, it didn't get slick or hard and made for oddly rather nice footing! however - be careful. check the footing and think if you can't jog on it, neither should your horse. if you're even questioning it, then don't do it because it can most definitely cause injuries (just like any other slick footing).
good luck! and barefoot barefoot barefoot! my horse goes in shoes w/cleats all around bc we hunt and pace and do jumpers and he get significantly less t/o and more riding work in the winter if we have a steady show schedule and i can't pull his shoes! cleats can offer too much traction and if your horse is prone to doing anything silly (like my jumper is), too much traction can quickly cause serious injuries to tendons and such. just something to keep in mind.