As many have pointed out ..... your problem are beavers and not muskrats.
I have been dancing with the beavers now for over 15 years ..... here are the lessons I learned:
1.) Beaver meat is really good if prepared correctly. There is a reason why natives preferred beaver meat to moose, deer, bear ... etc. .... One needs to remove the scent and oil glands cleanly and to avoid the contamination of the meat ....
2.) Trapping with Conibear traps is the most effective way to get rid of the beavers. However, you need to know what you are doing .... You can't just put up a trap and expect beavers to walk into it. And beavers learn fast! Very very fast!! The first ones of a colony are relatively easy to trap ... and as time passes and the beavers learn and become more and more suspicious .... it becomes more and more difficult to trap them ... and to the point of being nearly impossible to get the last ones of a colony into a trap. Sometimes the last ones need to be shot. (And yes, from time to time I am catching a muskrat in my beaver setups.)
3.) I am having some success with putting drainage pipes into beaver dams (and my creek) and to keep the water level at a certain level and to keep the beavers from raising the water level further. I am putting 10 to 20 meter sections of drainage pipe into the ponds .... and they work best if the drainage pipes are suspended somewhere below the water level and above the pond's bottom. Thats when the beavers have a hard time to plug those drainage pipes. Some of those drainage pipe installations have survived the beaver's determination now for over ten years ..... others, the beavers managed to clog within 3 -5 years and I had to redo those installations. I am still dancing with the beavers ... learning myself ... and perfecting that system .....
In the meantime .... I am enjoying my beaver BBQs ..... yummmiiii ......
p.s.: Trapping beavers (or shooting beavers) is a temporary solution. At one point I had trapped every beaver on my land and on my neighbors lands and in a radius of several miles in all directions. My land was free of beavers for 2 years .... and until the beavers had migrated back in on the neighbors lands and eventually back onto my land as well. Today, I am still trapping beavers and to control their population ... but I am focusing more on containing them and the damage they can do .... and by putting drainage pipes into the ponds and dams ... I have given up on the idea of getting rid of them completely.