I can tell you the complete answer for $g \leq 2$:

- When $g=0$ and $n \geq 3$ no nontrivial automorphic forms appear.
- When $g=1$ and $n \geq 1$ the class of automorphic forms which appear are exactly the cusp forms for $\mathrm{SL}(2,\mathbf Z)$. (Deligne, Birch)
- When $g=2$ and $n \geq 0$ the automorphic forms are precisely the cusp forms for $\mathrm{SL}(2,\mathbf Z)$ and the vector-valued Siegel cusp forms for $\mathrm{Sp}(4,\mathbf Z)$.

(In none of the cases will it make a difference for the answer above whether you work on the open moduli space or its compactification.) The $g=2$ story is spread across several of my papers e.g. arXiv:1310.7369 and arXiv:1310.2508.

One general statement you can make is that since $\overline M_{g,n}$ is a smooth proper stack over $\mathbf Z$, and $M_{g,n}$ is the complement of a normal crossing divisor over $\mathbf Z$, there is a strong condition on the automorphic forms involved: they must have "conductor one". Or, the $\ell$-adic Galois representations are unramified everywhere and crystalline at $\ell$. This is why in $g=1$ you only get cusp forms for the full modular group and never a congruence subgroup, and again for $g=2$ you only get the full group $\mathrm{Sp}(4,\mathbf Z)$.

There has been great progress in the past five years or so on the problem of enumerating automorphic representations of conductor one, by Chenevier, Renard, Taïbi, Lannes; perhaps others should be mentioned here as well. They can now tell you unconditional answers to things like "Up to some given dimension and weight, what is the complete list of automorphic representations of conductor one? What are their Hodge numbers?

There is a long-running project in experimental mathematics by Bergström, Faber and Van der Geer trying to study the automorphic forms appearing for $g=3$ and $n \geq 0$ by counting points on these moduli spaces over finite fields and interpreting the results. Unfortunately most of this work is still unpublished. They find that the class of automorphic forms they find is strictly larger than the class of Siegel cusp forms for $\mathrm{Sp}(2g,\mathbf Z)$ for $g=1,2,3$. Some of the "new" Galois representations they find should be related to Ichikawa's *Teichmüller modular forms*, but I'm not sure that they expect all the Galois representations to be accounted for in this way. This is based on things like them reaching $n=17$ (IIRC) where they expected to find for the first time a class in Betti cohomology given by a Teichmüller modular form. And then in the point counts they find some unknown six-dimensional Galois representations for exactly this $n$, transforming according to the right representation of the symmetric group $S_{17}$. So they expect this to be a six-dimensional "motive" "attached" to this Teichmüller modular form. And then they look in the tables of Chenevier et al and find that for precisely this dimension and weight they should actually find a nontrivial automorphic representation of conductor one (maybe it was a representation of $\mathrm{SO}(7)$ in this case?) and its Hodge numbers match up beautifully with various restrictions coming from geometry. In particular, assuming it all matches up they now have computed lots of traces of Frobenius on the Galois representation which should be attached to this automorphic representation which was just abstractly known to exist. And then they have repeated this for higher values of $n$. But from what they say they don't seem to find many patterns and I don't think they have any real conjecture of what the general picture should look like.

So already the $g=3$ case is a mystery and the higher genus situation even moreso.