There are many alternative compactifications of $M_{g,n}$ which live naturally under the classical Deligne-Mumford compactification $\overline{M}_{g,n}$. For instace the moduli spaces of weighted curves $\overline{M}_{g,A[n]}$ where $A=(a_{1},...,a_{n})$ is a vector of rational weights $0<a_{i}\leq 1$. These spaces were introduced in http://arxiv.org/abs/math/0205009.

Starting with weights $A = (1,...,1)$ by lowering them to $A=(a_{1},...,a_{n})$ we get a reduction morphism $$\rho:\overline{M}_{g,n}\rightarrow\overline{M}_{g,A[n]}.$$ The morphism $\rho$ is birational and in general it contracts some boundary divisors. Other moduli space, like moduli of Prym $\overline{R}_{g,n}^{r}$ and of Spin curves $\overline{S}_{g,n}^{r}$, admit natural forgetful morphisms over $\overline{M}_{g,n}$ which are finite and not birational.

**Are there moduli spaces (perhaps moduli of curves with some addictional structure) admitting modular birational morphisms to $\overline{M}_{g,n}$ ?**

In some sense a version of moduli of weighted curves birationally living above $\overline{M}_{g,n}$.