I have two dogs that like to run like crazy and when they run they also stop on a dime and cut my grass. It is down to the bare bones dirt and it is underneath a huge maple tree which gets no sunlight at all. I live in Michigan and my soil is mostly a sand base so it's not ideal for growing grass either, but what is the best type to use?
Visit the garden center. They have special grass seed for shady areas and for areas that have high traffic.
You can also call your local county agriculture agent and ask about what type of grass will grow best in your area (and he is not selling grass seeds or plugs). You will probably have to put something up to deter your dogs away from that area - at least until the grass takes a good foot hold.
If it's shady and has poor drainage you're wasting your time trying to grow grass. In addition to the shade, maple trees have a lot of surface feeder roots that will draw most of the moisture and nutrients out of the soil before turfgrasses can benefit.
But you can try if you wish, of course, according to this guide www.gardeningknowhow.com/
I'm in a similar position, and I have a tree that shades the front lawn. The sun doesn't get there at all in the winter and is only on the grass for a couple of hours in the evening. We've just put some artificial turf (wisepick.org/
It also works if you are a dog owner: pet's urine kills the grass, and astro turf is much easier to maintain.