Icicles hanging over the edges of roofs might look pretty, but once enough melted water gets trapped behind that ice, water can leak in to the house. Minnesota saw a ridiculous amount of ice dams two and three winters ago, which prompted many homeowners to start looking for ways to prevent ice dams.
The two main driving factors behind ice dams are attic air leaks and insufficient insulation. If those two are fixed, the potential for ice dams is quite minimal. A third component to ice dams is attic ventilation. Attic ventilation is required by code, but it's not an ice dam elixir. As long as attic air leaks are sealed and insulation is sufficient, only small amount of ventilation are needed to keep the attic space cold.
Unfortunately, lots of people claim that inadequate attic ventilation causes ice dams, which leads people to do all kinds of crazy stuff to get more air flowing in to their attic.
While the photo above of a desk fan installed in someone's attic is a little silly, the person who installed it obviously thought it was a good idea. There are actually fans made specifically for attics. Most need to be connected to a 120 volt power source, but there are also solar powered fans available, such as the one pictured below. This one was sold at Costco for a while.
My advice is to skip these fans entirely. While attic fans will certainly suck air out of your attic, the problem is that this air needs to be replaced. Where does the air come from? Some of it will come from the soffit vents, and some will come from attic air leaks. If you have problems with ice dams, a LOT of air will come from attic air leaks. This means you'll actually be sucking warm air IN to your attic.
If you have problems with ice dams, focus on fixing your air leaks and insufficient insulation. Once your attic lid is perfectly air sealed and you have enough insulation, the main causes of ice dams will have already been corrected. Don't waste your money on an attic fan. If you already have an attic fan installed, your best option would be to have it removed and have proper passive ventilation added if not already present.
What about using these fans during the summer? Nope. Check out any of the links below for more info on why you're better off not using attic fans... ever.
Related links, mostly pertaining to attic fans running during the summer:
Reuben Saltzman, Structure Tech Home Inspections