There are a lot of reasons and ways for water to get in but 20 inches of snow in areas where that is not normal will help explain some of them. With a little looking around with a flashlight, you can probably rule out many possible areas of concern since this will probably be as bad as it will normally get. Things like ice damming on roofs show up in funny ways as water runs through your walls.
Since your ground may be frozen pretty well, there is not as much to contain any runoff or snow melt. This is more probable. If you have a relatively new cast concrete basement, you might have some leakage around the rod locations where the forms for pouring the concrete. Those are the little holes. I have an older home and my kids have OLD houses, so I'm not really up on dealing with those. I'm sure that others here can guide you on that.