From the wiki page for piste maps at http://wiki.openstreetmap.org/wiki/Proposed_features/Piste_Maps I believe that you should tag the snowshoe trails as piste:type=hike, piste:grooming=backcountry
For cross country skiing it would be piste:type=nordic. That wiki page does not cover snowmobiles, and I never mapped for them (banned in areas I cross country ski) so I can't help there.
The wiki is silent on multiple use trails but I think the usual semi-colon separator could be used to indicate the different uses, e.g. piste:type=hike;nordic