Excerpt from an article by The Tire Rack...
While deep snow and ice-covered roads are two of the most challenging conditions North American drivers will face, tire developments during the last decade have noticeably advanced wintertime mobility. The technological revolution of dedicated winter tires for drivers in the snowbelt, and the continuing evolution of all-season tires for drivers living on its periphery characteristically offer more grip in snow and on ice than ever before.
In most parts of the world, tires are considered to be legally worn out when they reach 2/32" (approximately 1.6mm) of remaining tread depth. U.S. law requires tires to have easy-to-see Tread Wear Indicator bars running from one side of their tread design to the other when the tire's tread has worn down to the minimum legal limit of 2/32 inch.
The Tire Rack recommends that drivers expecting to encounter snow-covered roads consider replacing their tires when they reach approximately 6/32" of remaining tread depth to maintain good mobility. Tires need more tread depth in wintry conditions to compress snow in their grooves and release it as they roll. If there isn't sufficient tread depth, the "bites" of snow that can be processed on each tire revolution will be reduced to "nibbles," and the vehicle's traction and mobility in snow will be reduced.
You can shop for studless ice and snow tires here, and performance winter tires here.