It's been over a decade since I went, but I went there (and Bryce Canyon) twice during the winter. You practically have the whole park to yourself!

With Bryce Canyon, it's best to go before snowfall, though. Once the ice sets in, they shut down a lot of the trails.

Also went once to Arches in winter. Awesome (except again, a lot of trails too slippery).

Yeah. I went through Bryce in the winter and you couldn't do a lot of hiking below the rim. The trails just weren't signposted or set up for it. Beautiful from the rim with the snow but the canyon itself wasn't really accessible.