Icicles may look pretty and seasonally appropriate, but they're also a sign that heat is escaping through your roof and causing the snow cover to melt. As the melting snow refreezes, it can block the gutters and keep water from draining away as it should. Then, when the ice and snow melted for good, water may have seeped into your walls or dripped onto your house's foundation instead. That can lead to many different problems.
And, next time it snows, consider using a broom to brush away the snow from lowest few feet of your roof. You may feel silly being out in the cold "sweeping" your roof, but giving the snow melt room to drain unobstructed can help you avoid costly problems with mold, a cracked foundation, and damaged soffits, fascia boards and walls.
To prevent ice dams from occurring, you may need to improve the insulation and ventilation of your roof and/or install a heat strip along the lower edge of the roof or in the gutter. For now, sweeping the snow away will help the situation. And then, when it warms up a little, you can consider your other options.