I had the same problem with a previous garden, although it did get some sun, but the grass didn't. I got rid of it altogether and replaced it with gravel and beds. I had a stepping stone path too. There are lots of shade loving plants.
I wouldn't waste my time on grass in that situation, the alternative looks great, and a bench amongst the plants works well too.

I agree with Mad wet sunless areas are not good for grass - moss is a different thing :-)

we get quite alot of boggy areas but my hubby put extra drainage pipes in to take the water, luckerly most of my grass does get sun, its my patio thats the worse area