Experts and BHG readers answer.
Moles in the snow
The trails that you see as the snow melts are most likely those of voles, a mouse relative that tunnels under the snow through the grass in search of seeds and other food. Keeping areas mowed closely is one of the best ways to keep the vole population low, though they do go through population cycles.
If you see raised ridges of soil being pushed up, that is more likely mole damage. They are not usually active near the soil surface during winter, but if snow cover has been heavy, the ground may never have frozen, and they may come up near the surface.