This survey paper from Lizhen Ji says:
Teichmüller was aware that nontrivial automorphisms of Riemann surfaces caused difficulty in constructing $M_g$ and singularities of $M_g$, and he introduced the idea of marking to rigidify Riemann surfaces and hence to kill nontrivial automorphisms of Riemann surfaces. This is the meaning of topological determination.
I wonder what the singularities of $M_g$ mean? In my mind $M_1$ is biholomorphic to $\mathbb{C}$, does that mean there are no singularities of $M_g$? Are there any examples?