That's an excellent question, especially for those of us in colder climates. From personal experience and talking to manufacturers, the performance really hinges on a few key factors.
First, material is everything. Heavy-duty plastics like polyethylene can become brittle and crack in extreme cold if they're not specifically formulated for it. Look for cans labeled as "all-weather" or "frost-resistant." Metal cans, particularly galvanized steel, are generally more resilient against cracking from temperature alone, but they can be prone to rust if the coating is damaged and moisture gets in.
The lid mechanism is the most common point of failure. Hinges, especially plastic ones, can snap when you try to open them if they're frozen shut. A simple, robust design with metal hinges or a one-piece lid is best. Also, a tight seal that keeps snow out in a blizzard can also freeze into place. I've found that a quick spray of silicone lubricant on the seal and hinges in late fall works wonders to prevent ice adhesion.
Another issue is waste freezing to the inside. Lining the can with a bag helps, but sometimes the bag itself freezes in place. Tapping the sides of the can with your foot before emptying can often break the frozen mass loose. For permanent installations, some people opt for cans with a slight taper, which makes it harder for frozen contents to get stuck.
Maintenance is straightforward: avoid forcing frozen parts, clear heavy snow accumulation off the lid to prevent strain, and if possible, position the can in a spot that gets some winter sun. A high-quality can designed for four-season use, with a bit of preventative care, will absolutely hold up through winter after winter. The cheap, thin plastic ones, however, are often a casualty of the first deep freeze.