This past weekend had me out on the mountain bike both days - a short ride Saturday followed by maybe 10 miles or so on Sunday. Saturday I tried the bike path again, but I think that deep snow riding is something that I'll mostly avoid. Once the snow has sat for a few days, it settles, hardens, and develops a crust, all making it that much more resistant to the passage of a bike tire. It gets to be brutally hard work just to keep moving forwards, let along covering any distance. When it's fresh, light, and fluffy, it's a hoot, but that doesn't last, especially once some wind comes up.
Sunday I stuck to the roads, and road-style riding, avoiding the snow as much as possible. It's still hard work, given that I'm bundled up, on my mountain bike, and on fat knobby tires inflated to the bare minimum for the sake of as big a contact patch as possible. On the hard surface, though, I can keep up a decent cadence in a middle gear, and feel like I'm actually getting somewhere in reward for my effort. Cars have generally been good; I think a lot of drivers rightly regard someone riding in this weather as an oddity. When the road allows, I do duck over onto the shoulder to give myself more room.
I'm getting the clothing layup closer to right, too. I think a three-layer approach is best: The first, innermost layer is primarily devoted to moisture transport; i.e. getting sweat away from my skin. I've been wearing bike shorts first (for the chamois), then either my leg warmers and a pair of old running tights, or some polypropylene long underwear. Polypro goes on top, too, possibly over my arm warmers.
The middle layer is devoted primarily to insulation; it's synthetic fleece that's the obvious choice there. I've got a nice selection, but either a zip-up vest (if I'm wearing the arm warmers) or a long-sleeve pullover are the basic choices up top. On my legs, if it's not truly cold, I won't add a middle layer, or at most a second pair of tights. I do have a pair of fleece sweats that would work fine in really cold conditions.
The outer layer is first there to block the wind, then to keep water from coming in and to allow sweat to dry outwards. I have a cheap nylon windbreaker set that I got from West Marine years ago; the pants seem ideal. The top is OK, but I've only worn it once so I'm not sure. This weekend I wore the vinyl rain jacket I got last fall; it's got mesh vents that are supposed to let vapor out, but I still look like a greenhouse after a few miles so I think the vents aren't enough. I'll go back to the nylon and see how that works.
I know I've got it close to right when, after a ride, the inner layer is dry, and the middle layer is full of sweat, and the outer layer looks wet.
Oh, on my hands, I've worn deerskin gloves, which were OK but let too much cold in though the cuff, and heavy Gore-Tex ski mittens, which were too hot for mid-20's weather. I need to work on my feet; two pairs of socks don't add a thing. My tootsies had no feeling when I got home Sunday, so I warmed them up for a few minutes in the bathtub. Some folks swear by a plastic bag over the socks inside the shoe; otherwise, I'll look for overbooties.
My mountain bike has done fine. The shifting has never frozen, although it has gotten sluggish and I've been frozen out of some of the faster gears. I've been spraying it with a aerosol wax lubricant (I think it was Elmer's Slide-All), which has helped a bit with snow buildup. The cheapo tires that came with the bike are doing fine, much to my surprise. People recommend studded tires, but since I'm mostly on either snow or pavement, I don't know how much they add. At $100 a tire for good Nokians, I'll live with not knowing. Maybe I'll try making my own; there are instructions on Icebike