For snow (and ice) - get Schwalbe Marathon Winter tyres, 35-622 dimension. That's what I'd recommend.

For rain and cold weather if a material you ride on is harder than the tyre (concrete, even poor one), slick tyre is the best tyre profile. Just make sure the rubber compound is soft and the tyre is not too narrow. If material you ride on enables the tyre to dig in (mud, grass, even gravel), then tyres with more aggressive tread pattern are better.

I wrote article on slick tyres (and tyre grip in general):
Slick tyres - Cycle Gremlin

And an article on winter tyre choice, according to your winter riding conditions:
Bicycle winter tyres


If you have such an on-and off winters, a good choice would be having a second winter bike with set up studded tyres. That's what I do.
If you live in flat lands and roads are salted and ploughed regularly, you could also "get away" with just swapping the front tyre - or having a spare front wheel with a mounted studded tyre. Front one is the most important for staying upright.
Or, to be safe, you could put a front studded tyre for the whole winter and then swap the rear wheel - studs-no studs, according to weather. Then you'd have two studded tyres in the snow, to be as safe as possible, but won't wear out studs too fast riding rear studs on dry pavement. Though front wheel will make some noise on dry pavement and slightly worse traction on bare concrete than a non-studded tyre.