Fenders and mudflaps are essential to keep you and your drivetrain clean.
As for tires, it's personal.
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Studded tires will allow you to cycle more aggressively without having to worry about ice ruts, tramway tracks and black ice. On the other hand they weight a lot (i.e. good training) and are a bit noisy. Don't get anything but Nokians because all others use steel studs that wear down very quickly on asphalt. The "low stud count" Hakkapelliitas are plenty for commuting.
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Smooth-rubber knobbies are quite good in snow or sleet, but you have to ride them more carefully if you fear there is ice on the road.
The ideal solution also depends on the distance you travel and whether or not you want to walk or take the bus on nasty days. Other options include:
– Studs in front and knobby on the rear wheel, giving you less drag and most safety benefits.
– Knobbies on the front and rear wheels, with an extra wheel equipped with a studded tire. That way, you ride most of the time with knobbies, but you switch the front wheel when snow or ice is expected.
I'm in Montréal. I have commuted for many years using Specialized Cyclocross knobbies (700x37) and Continental Mount&Ground. Last year I had Nokian Hakkapelliittas on my commuter bike (used every day, but mostly for distances lower than 10 km) and knobbies on my touring bike (used for longer distances... but not when the weather is nasty).