We eliminated our Beaver problem (for now!) when I had a pair of trappers come in and take 4 of them out this time last year. The largest was 60 pounds! They were doing a real number on the trees in our property and their dams threatened to flood us out this past Spring.
We have a constant problem with Muskrats and Mink. I saw 8 Mink all at the same time while cutting grass this summer. They come into our pond from a nearby river. I am currently trapping the Mink and got two so far this week. They are doing some real damage to the fish population. They leave their scales and half eaten prey all over in small piles and run around during the day fishing at their leisure!

I would have to be very quick and accurate to shoot them.