I didnt want to break the bank when i purchased different tires for my sportsman 500. I found some kenda bearclaws and really liked them. (even went a size up to give me more ground clearance) Worked great in the snow and in the mud if needed. They are wearing nicely (make trips to the hills every year to ride trails) have over 2000 miles on them. wouldnt hesitate to put them on again for the price.
If you want to stick with atv style tires, I can attest the ITP Blackwater's that we put on the SxS have been pretty impressive. wouldn't call them a great ice tire, but mud, sand, nasty rock and snow they do well. They've also been impressively tough. We ride down in the black hills and although my rims too a beating from rock climbing trails , the tires had zero issues and you couldn't tell from any wear on the tires we had been climbing up some gnarly jagged rock trails. they also have good stiff sidewalls unlike a lot of atv tires.