By the pointed canonical bundle formula the canonical bundle of $\overline{M}\_{g,n}$ is given by
$$K\_{\overline{M}\_{g,n}} = 13\lambda+\psi-2\delta-\sum\_{I}\delta\_{1,I}$$
where $\lambda$ is the Hodge class, $\psi = \sum\_{i=1}^{n}\psi\_{i}$ is the sum of the psi-classes and $\delta$ is total boundary divisor.


**Are there constraints involving $g,n$ which imply that $K\_{\overline{M}\_{g,n}}$ is ample ?**



Canonical bundle of the moduli space of curves
The cotangent bundle of $\overline M\_{g,n}$ is *never* ample. You can see this for instance by restricting it to the hyperelliptic locus. 


There has been a lot of work in the last 5-10 years on running the (log) minimal model program on the moduli space of curves. You can read about this to get a feeling for how far from true it is that $K\_{\overline M\_{g,n}}$ is ample. There's a survey of Fedorchuk-Smyth in the Handbook of Moduli, for instance.


