Around here ice is rarely an issue, and the roads are heavily salted enough that it's typically slush and not snow that you're pedaling through. I've done two years quite comfortably on 700C x 32 mm tires. The ones with a little tread on them work better in heavy snow . . . bald tires just spin in my experience (although are fine for pretty much all other road conditions).
For icy conditions, you can make due by radically altering your cycling habits, but really need studded winter tires for control. Nothing else will grip at all.
Cycling on ice is similar to cycling in deep slush and packed snow . . .
- slow down
- don't lean the bike much when turning
- don't turn and brake
- don't turn and accelerate
- when braking, only brake in a straight line
- leave plenty of room in front of you to brake
- Use your front brake sparingly (I use it alone about 90% of the time . . . but it's more like 30-70 front to back when very slippery) because if your front wheel starts to slide you're going to wipe out
- use flat pedals (I like studded ones that grip your boots really well) so you can put a foot down quickly if you start to fall
- put your seat a bit lower than you normally would so you can get a food down quickly if you start to fall
- shift gears often (helps to prevent ice from forming on your cables/derailleur and getting you stuck in one gear)
- brake early (you need to clear the crap off the rims before your brakes start to work) and often (same issue as with the shifters, sometimes the brake cables get frozen in place if you don't work them often).
- really get to know your bike route. This way you can anticipate where the icy spots will be and plan your speed well ahead.
If your city uses as much salt as we do in the winter, you need a VERY regular cleaning schedule for the bike. (Like daily rinse off and lube, bi-weekly checking the brake pads and pivots, greasing every screw that goes into the bike before the winter, using some waterproof grease in the wheel bearings, etc.) It really is a lot of fun to bike in the winter despite all the extra work it ends up being!