A [related MO question](https://mathoverflow.net/questions/38968/betti-numbers-of-moduli-spaces-of-smooth-riemann-surfaces) asks about Betti numbers of $M_{g,n}$ and has some quite complete answers. However, it focuses on the case where the $n$ marked points cannot be permuted.

(Sorry, I know this should have been posted as a comment, but I do not have enough karma yet.)