For a striaght Ice and Snow winter tire that is still good in mud and on dry pavement, the Nokian Hakipilaten are excellent tires. I don't like changing my tires summer and winter so I run Goodyear Duratracs year round. They are sold by the millions to companies that spend a lot of time on gravel and mud roads, like oil service companies. They are a great all round tire, not the ultimate on ice, but very competent. They carry the snow symbol so are approved for use without chains in the Canadian Rockies during winter. If you look up their rating you will see they outsell every other similar tire ten to one.
For those of you that do serious mud, BFG Mud Terrains that have been heavily siped by a truck tire shop are also an excellent choice. Has to be done when the tires are brand new, once you have rocks in them or any use no shop will touch them for siping. Tremendous mud traction and great on ice and snow.