Mice are the light, fast scratchers heard at night; rats are heavier and tend to be lower in the house, near water and the foundation; squirrels are daytime animals, mostly early morning and late afternoon, and they are up high, in the attic and soffits. If the noise is at 6 a.m. in the ceiling, think squirrel. If it is 11 p.m. in a kitchen wall, think mouse.
On Long Island we see two mice indoors. The house mouse is gray all over, small, and lives inside year round. The white-footed mouse and its cousin the deer mouse are brown on top with a clean white belly and white feet, and the USDA and University of Nebraska wildlife damage handbook notes they are uncommon in dense suburbs unless there is open space nearby, which is exactly the North Shore, the pine barrens edge in Suffolk, and anywhere backing woods or a park. Those outdoor mice push in when it turns cold, so a wall that was quiet all summer in Oyster Bay or Smithtown can start scratching the first week of November.
The treatment is the same for all of them: find the entry, trap along the runway, seal the gap. The difference is where you look. House mice are usually low, coming in at the sill plate, the garage door corners, and the AC line set. White-footed mice are good climbers and will use a tree branch touching the roof or the gap where the electric service enters the house. If you want the full identification breakdown, our guide to scratching noises in the walls at night covers mice, rats, squirrels, and raccoons side by side.