>>381577A 'cross bike should handle 2-3" of snow just fine (and also moderate logs and rocks, if you actually learn trail riding skills instead of relying on suspension). It depends on how wet, dense or lumpy the snow is but narrower tires can actually cut through better.
>>381540Cyclocross bikes are great for that kind of stuff. I ride my cross-check all year on paved roads, gravel roads and some easy to moderate mountain bike trails (unless it's too muddy, then my MTB with 2.1" knobbies does better). As for snow, with 35-38mm tires it does pretty good in stuff like picture related. I've also got studded tires (Schwalbe Marathon Winter 35s) for black ice and riding on the lake ice.
