I commute by bike through winter, at least until ski season is in full swing, then I'll slack off a bit. My commute takes 45 minutes to an hour and a half, depending on the route.
My commute is no problem at all down to about -5 C/23F. It's easy to dress for and the bike works just fine. I need studded tires once it gets icy, of course.
Down to about -10 C/14 F it can still be fine, but the bike starts protesting. Gears are more sluggish to shift. I have to wear yet another layer and think about how much cold air I pull in.
Down to -15 C/5 F can be a lot of fun to ride in, but it's not really that cycle-friendly. I usually have to settle for maybe half my gears. The air is dry, which is nice, but it's cold enough that if you're riding fast enough to break a sweat, having to stop for any reason is not much fun. Forget fixing a flat, or anything else that needs bare fingers. But biking to work is still warmer than hanging around waiting for a bus 
Below -15 C I'll only bike very rarely, it's a bit too much of a hassle. Keeping feet warm is the hardest.
But in my experience snow is more of a dealbreaker than cold. Cold you can dress out, but snow you have to ride through. Falling, wet, heavy snow is bad, piles up and clogs everything, at the same time drivers are skidding every which way. Light dry snow is no problem.
Hybrids are quite common here and you can get studded tires for them, but most winter commuters seem to prefer mtb's. Maybe because they're more versatile and nimble when it's slippery?
PS. The only real mechanical problem I have regularly in winter is with plastic. Anything in plastic gets brittle and breaks easily. You also have to reckon on a lot more wear on the drivetrain. I've been told one winter can equal the wear of four summers, and I agree.
Winter riding is much less about badassery and much more about bundle-uppery. - malkin
1995 Kona Cinder Cone commuterFrankenbike/Selle Italia SLR Lady Gel Flow
2008 white Nakamura Summit Custom mtb/Terry Falcon X
2000 Schwinn Fastback Comp road bike/Specialized Jett