In general, you want a drivetrain with low gears, the ability to fit snow tires (and maybe fenders), and a bike that's agile and maneuverable yet able to stay upright and steady with little effort.
What tires are good for snow? Wide tires are good, knobbies are good, but studded tires will help you on ice and packed snow (and maybe on fresh but wet snow, although that's debatable).
Here's my run-through on various types of bikes and how they'd fare as snow bikes:
Road bikes generally take skinny tires and are unsuitable for ice and snow, although there are exceptions. The drivetrain will have those low gears, mostly, but these are built for speed, not slow riding. Road drivetrains are relatively fragile and would be more likely to be harmed by road salt and ice.
Touring bikes can be fitted with snow tires, and they usually have room for fenders and those studded tires (although they may get an accumulation of snow packed in between the tire and fender). They're quite sturdy and can take a beating, and the drivetrain will handle road salt well as long as you clean it from time to time, but you get what you pay for: a food steel touring bike is heavy and expensive. These will have clearance to mount fenders. Touring drivetrains are built with low, low gears to climb hills with a load of camping and cooking stuff - and low gears like this are great for traction on ice.
Folding bikes are amazingly useful in the city, as they pack up small (think: storage in a city apartment, and taking them on city transit). Small wheels can be hard to find studded tires for, but are amazingly nimble. My Dahon Curve with 16" knobby tires does a respectable job in the snow. Downside: A bit more expensive that a bike of comparable quality, as they're more complicated machines. Because, well, they fold. An upside is that these bikes often come with internal hubs, which are amazingly resistant to snow and road salt.
Mountain bikes are built for trails and dirt, come with 26" tires - easy to find snow tires for, and stock knobbies can already handle light snow and ice fairly well. There's usually room for fenders as well. Mountain drivetrains will have the low gears you want for traction on slippery surfaces. Whether the drivetrain can take a winter depends on the bike, but most should be comparable to a touring bike.
BMX bikes - Not sure if you can fit fenders on these, but I'd guess that's touch-and-go. 20" studded tires aren't unheard of. Would make a good snow bike if you can find tires and fenders for it. My experience with small-wheeled bikes leads me to guess that these might be a little twitchy on snow - but the low center of gravity might compensate for that. Anyone have a comment on this?
Cruisers are kinda like mountain bikes with a long wheelbase, a more upright and comfortable position, and they're not build for speed or hill-climbing. Unsuitable for snow or ice, unless you live in a very flat area and are okay with going very slow. Cruisers often have internal hubs or even a single-speed drivetrain, great for dealing with snow and road salt.
In the end, unless it has a very delicate drivetrain I'd go for using the mountain bike for snow and ice. These bikes are already built for rough conditions and, importantly, you can get 26" studded tires fairly simply. BMX bikes usually take 20" tires, which are harder (but not impossible) to find studded tires for.
If you want to keep your new bikes in better condition and will buy a bike specifically for winter riding - hardly unreasonable, many riders do this since winter riding is hard on a drivetrain - consider a used steel or aluminum hardtail mountain bike. The hardtail frame will give you incredible control. There's a question about flat bars vs. riser bars (as well as stem length) right here. (That bike will be used as a winter bike as well as a bike for gravel trails in the summer.)