If you have persistent ice dams, and other solutions are difficult to implement, don't hesitate to use deicing cables. Ice dams result from repeated cycles of thawing and freezing, which cause snow and ice to melt, then freeze along the roof edge (lead photo). This “dam” traps water, allowing it to work under the shingles. Deicing cables heat the roof edge and keep it free of ice.
Install the cables along the roof edge where ice tends to build up. Attach the cables following the manufacturer's directions. Then simply plug them into a ground fault circuit interrupter receptacle after heavy snowfalls or when you see ice dams beginning to form. These cables are available at select hardware stores and home centers, starting at $35. Systems that include sensors that turn on automatically cost substantially more. Also find them online. (One brand is Easy Heat.)
The only downsides are the cost of electricity (which is small for a standard cable used intermittently), and routing the draining water away so it doesn't freeze and build up at the end of the downspout.