In the Boston area, where I live and bicycle, the most dangerous winter conditions are:

Snow banks on the sides of the roads make the roads narrower, squeezing bikes more into the traffic lanes.

Out-of-control, skidding cars are a real, all-to-common hazard.

I usually try to stick to the bike paths to avoid the above-mentioned winter hazards, but I've wiped more than once on ice. Bikes are much harder to handle on ice than cars, since they slide sideways out from under you if you alter your weight balance or direction even minutely. Luckily, if you're on a bike path, you won't be run over by a car the moment you go down (see number 2 above). The other disadvantage of the bike paths I use to get to work is that they are not plowed. People walk on them, then the slush freezes, making extreme washboarding that can just about vibrate the teeth out of your head.

It's a tough ride either way (paths or roads) once the snow starts to fly.

According to Hansen and the IPCC, winters will have disappeared by 2009.

Then you can bike in Boston in December in koolats.

Just wait one more year...then all the Global Warming doomsday scenarios will surely come true.

i ride to work year round and find the winter conditions challenging but fun.  i choose to use a single speed mountain bike with studded tires slightly deflated.  they handle fantastic and if you only ride that bike when conditions call for it they last many seasons.  also having fenders is a must in my opinion.  also be ready for some increased maintenance as you will be gunking everything up when riding through the sludge.  specifically clean your chain regularly and grease it.  also if yoiu have aluminum parts beware and make sure you wash them regular if you are in an area where they use salt.  as far as clothes i think the winter time is the perfect time for bib tight.  seals out the wind and grab a balaclava to seal the neck and head.  i ride in CT and have seen it all. the worst is the cars. nothing we can do about them though.  stay safe and happy winter riding.

... training wheels? They would give you that extra level of stability in the snow.

Anyone tried that?

That and watch out for snow plows.  I quit biking in the snowiest part of winter last year because I got hit by the snow flying off the snow plow.   When it gets too snowy, I switch to x country skis.  I picked up a used pair for twenty five bucks.  

This is my first year commuting via bike. The biggest trick is staying cool while staying warm. I have found that a pair of bike tights rated for cold temperatures have been great. The high tech fabric does a great job keeping my legs at the right temperature, wicking moisture while retaining heat (the right amount of heat). The few days I have not worn them, I came close to overheating. They cost about $100, but they will keep giving back by allowing me to ride into the winter. It was pretty easy to swallow that expense when I thought about it that way.

Anyone have advice for just plain cold?  We don't get much snow here in TN, but we do get cold and windy.  
And what about keeping the hands warm... wind goes right through my knit gloves...all the way to the bone!

Snow is OK, but ice can be deadly.

Last year, I managed to fracture two ribs as the result of riding across a narrow little sliver of ice on the pavement.

Whatever gear you choose to employ, be aware that winter riding seems to carry a significantly higher risk of injury. Partly, that is because of ice and adverse weather. Partly, it is because drivers will not be expecting cyclists as much.

This is second hand from my uncle Jerry, who biked year around for decades in Madison, WI. It was a great joy in his life.

First, replace the grease with oil to compensate for the cold. Second, never take a shortcut across the lake. Sure the traffic is lighter, the surface more consistant, but he fell in a pressure crack unexpectedly and nearly froze. (It happened in a heartbeat.) From then on he took his chances on the city streets. At least if he were hit, someone might notice!

I bike through the whole winter in Iowa.  On ice and every other condition.  I take it slow in the beginning of the season to get used to it, but by the end I am speeding over all conditions.  What I find the worst is riding down a fresh snow that has had only a few cars on it.  They pack the snow some what and then it breaks out from under you.  Not really dangerous just annoying, and difficult.  I just wear my normal clothes with one more layer.  I live in the country so the wind gets whipping out there riding home in the dark.  Many times I don't have to worry about getting hot.  

Hailing from Winnipeg, we get some serious winters to contend with, but it's my favourite season to bike in! You get to hold up traffic by riding in the ruts and you look like a maniac when you arrive at work/school covered in frost head to toe from your frozen sweat (whicked away from your body by umbra's advised non-cotton clothing). It's sweetass. My piece of advice is about tires. Personally, I like to ride slicks year round. I find they cut through mounds of snow/slush better, and snow doesn't stick to em as much, which means it doesn't get clogged in your brakes either! Plus it's badass. You just have to expect to fall sometimes. But it's the winter, and you're wearing plenty of layers aka padding. Happy biking!

We don't get the temperatures of Cleveland(or the snow amounts), but I have gained experience from year-round riding.
 On longer commutes, take a second pair of clothing...or at least gloves and socks.
 Dress to be slightly cold.  You will ride faster and warm up quicker.
 Fenders and lights, lights, lights.
 Take the lane.  None of this crap of riding in the ruts or on the snowy edge of the road.  
 When you throw your leg over the top-tube, remember to tell yourself that "people are nice."
I smile, sometimes wave, and generally have a great ride.  

Best of luck.  

... whatever you do, don't be stupid and pack your laptop with you (like I did) or anything else that's fragile.

Fenders and lights are a must.  Lube up more often.  If you have clipless pedals, consider changing back to the old fashioned regular pedals so that you don't break a leg when you fall on ice.  

Your front tire is the most important tire.  If the front can make it through something, chances are the rest of the bike will follow.  They now make tires that are made of a special compound that are a bit stickier.  They're called slow rebound tires.  I have a friend who took an old tire and put small nails through it so that the tire is covered with small metal studs.  Works for him!

The most efficient thing I've found to help keep me warm is a neck gaiter.  Anything to keep the neck warm really helps, so I suppose a scarf would work too, but I'm not that stylish :)

Also, I have not found much use myself for fancy tires, I think the biggest thing on ice is just staying balanced and going slowly... sometimes, really, really slowly.

I'll admit, a good pair of triple-layered gore-tex pants are worth their weight (which isn't much) in gold.  The pants breathe, but keep you very dry.  However, these pants can be pricey.

What I did was to go to a fabric store that stocks outdoor fabrics and bought swaths of double layer ultrex fabric and some light fleece for lining.  The double layer fabric is cheaper than the triple layer, and the fleece lining ensures that you will be comfy and warm inside.  I'm really not sure what temperature these pants will be rated for, but I live in Seattle and we really get wet; not cold.