Do there exist regular symmetric differential forms on $\overline{\mathcal{M}}_{g,n}$ the DM-stack of stable genus $g$ curves with $n$ marked points? By this, I mean nonzero sections $$ \omega \in H^0(\overline{\mathcal{M}}_{g,n}, \mathrm{Sym}^k \Omega^1_{\overline{\mathcal{M}}_{g,n}}) $$
It is shown here (or by vanishing results fro $H^1(\overline{\mathcal{M}}_{g,n})$) that there are no such forms in the case $k = 1$. I am wondering about the case $k > 1$.
This is not exactly what you are asking, but if you allow the symmetric differentials to have logarithmic poles along the boundary of the Deligne-Mumford compactification, the answer is "yes". This follows from the existence of interesting variations of Hodge structure on $\mathscr{M}_{g,n}$ by the main result of
Brunebarbe, Yohan. "Symmetric differentials and variations of Hodge structures." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018.743 (2018): 133-161.
For an example of the requisite interesting variations, you can consider $W^1R^1\pi_*\mathbb{Z}$, where $\pi: \mathscr{C}_{g,n}\to \mathscr{M}_{g,n}$ is the universal curve.
