I have spent the last couple of days looking for a 3D physics engine that plays nicely with MonoGame.
To save you some time, these are my results so far.
There are two versions, 1.1 seems like it may work with MonoGame, but it looks very slow for collision checking.
1.2 I cannot get to compile. Looks like it uses a new C# spec that is only available with pro versions of Visual Studio.
This was written for XNA and sadly uses a custom content processor to add physics variables to meshes. I tried working around that to make it more useful, but a simple cube kills performance. Spheres are fine you can have millions of them. In the end though I decided it was more trouble than it is worth.
I went through BulletSharp, couldn't get it to compile. However I found BulletXNA which someone wrote some time ago.
This also was written in a very strange way, though the class library itself looks ok. I have ported it to MonoGame and reworked all the demos so you can run them from a single windows app.
At the moment this looks the favorite for my use, though I am pretty sure there is a memory leak in it somewhere.
Anyway, to save you time if you are in the same position as I am, I have published the code on my website.
They are quick hacks just to test the physics, but you can grab and play with them if you wish.