During the winter, snow probably covered your roof and filled your gutters. But the potential danger to your gutters is when the snow begins to melt then refreeze when the temps drop below 32 degrees, as well as when temperatures remain above freezing and all that melting snow and ice need somewhere to go. If your attic isn't well insulated, melting snow can create ice dams which can seep into your home, causing damage to interior drywall and ceilings.
In addition to ice dams, ice built up in your gutters over the winter may have caused problems as well. If there was too much ice in your gutters, the added weight may have made the gutters so heavy that they actually ripped away from your home. If you clean your gutters after a heavy snow, this can help remove some of the added weight.
Leaking gutters are also an issue. Gutters full of ice and snow begin to melt, often overflowing over the edge of the gutters before it can make its way to the downspouts. This can cause damage to the areas below the gutters, including decking, landscaping, and erosion problems, as well as possible basement flooding.
If you have gutter guards or a gutter protection system, often the issues of ice and melting overflow can be lessened. If your gutter system has a topper, water that runs down your roof will flow over the top of the gutter system and away from your home. Without gutter protection, that water can sit in your gutters and possibly refreeze.
Now that most of the snow has melted, it's a good time to do a thorough cleaning of your gutters. Flush them out with a garden hose to clear out any debris before spring rains begin. Also shoot a direct water spray from the top of your downspout and ensure that the water flows out freely from the end of the downspout. Make sure that the water exiting your downspouts drains far enough away from the foundation of your home.
Spring is also a good time to make sure your attic is sufficiently insulated. As mentioned, ice dams aren't caused by gutter systems or roofs but rather heat loss through your attic. Properly insulating and ventilating your attic can prevent any ice dam problems next winter.