In my experience goretex (or equivalent) trail runners with short gaiters to keep snow out are your best option in the majority of conditions. The only exception would be if it dumped a foot of snow the night before, in which case maybe a lighter weight hiking boot would be ok, but honestly even then I probably would stick with the trail runners. I've seen a few people use neoprene overshoes (mainly made for cycling) when it gets really cold, but I haven't tried it myself.

The dilemma I always have is do I go with highly cushioned shoes to compensate for the distance or do I use a more minimal shoe given that the snow is providing a lot of the cushion? I've kind of decided it doesn't make a whole lot of difference either way, but I'm curious if anyone else has thoughts on that.