Holes in yard next to lake shore
1) Can you name the critter that is making these holes and undermining the whole area? 2) Do you know how to get rid of them or send them somewhere else? Picture 1 is the general area. Some of the holes are bigger, maybe 8" across. Others are smaller and more defined, 1.5" across (picture 2). Picture 3 is an area about 6 square feet and you can see how many holes they made in an area that small. It is very difficult to walk in the area because of the unevenness of the ground created by the holes. We want to put dirt on the area, level it off, and plant grass. But the critters will just make more holes unless we can figure out what to do about them. We have obviously tried putting rocks over the holes.
Genesee County Michigan
From your photos and description, my best guess is chipmunks, or potentially ground squirrels. The size of the holes seems right and the lack of piles of dirt outside the holes is a characteristic of both - they carry dirt away in the cheeks. Have you seen either around your property? With that many holes I would guess they are looking for food. You could lay hardware cloth down temporarily - or even bury it under some soil - to stop the burrowing. Your grass would grow right through the hardware cloth if you wanted to leave it there.
We also recommend consider deeper-rooted plants near lake shores (it's hard for me to tell from the photo how close this actually is to the water). Lawn grass roots aren't deep enough to prevent erosion if waves or ice hit the area, but taller plants can do the trick (and once established, are much less appealing to little digging rodents).
Muskrats are also definitely a culprit to consider, especially near the water. Since most of the holes look fairly small, that's why I was thinking chipmunks. But if there are also some larger holes (you mentioned up to 8") that could easily be a muskrat. You could have a few different species doing damage, actually.
Given your interest in erosion control options, I recommend you check out the Michigan Natural Shoreline Partnership website. This resource is maintained by folks from MSU Extension, the DNR, and others, to show ways to minimize erosion while keeping the lakeshore as a healthy habitat for desirable critters (like birds, fish, and frogs). You may find it interesting to check out! It includes nice lists of good plants with deep roots to plant along lakeshores.
Here's the link: http://www.mishorelinepartnership.org/
I have not personally worked on nuisance muskrat control. The Michigan DNR recommends contacting your local DNR wildlife biologist, or a permitted Wildlife Damage and Nuisance Control Business, for advice. You can find your DNR wildlife biologist by contacting your Customer Service Center (https://www.michigan.gov/dnr/0,4570,7-350-79137_79765_81257---,00.html). Here is the DNR's latest list of private businesses that have permits to appropriately address nuisance wildlife: https://www.michigan.gov/documents/dnr/WLD_Wildlife_Damage_and_Nuisance_Control_Permittees_366435_7.pdf
Finally, here's a fairly recent article from Michigan Lake Stewardship Associations with some "do it yourself" suggestions: https://www.mymlsa.org/muskrats-in-the-lake-and-in-your-boat/