For $g\geq 2$, let $M_g$ be the moduli space of genus $g$ hyperbolic surfaces, and let $\lambda_1(S_x): M_g \to \mathbb{R}$ be the smallest eigenvalue of the LaplaceBeltrami operator on the surface $S_x$ parametrized by a point $x\in M_g$. Is there anything known about how the values of $\lambda_1$ are distributed when viewed as a function on moduli space? For example, does the volume of the set of surfaces with $\lambda_1(S_x)<\varepsilon$ go to zero rapidly as $\varepsilon \to 0$?

Not anywhere near as much as known as one might like, but for enlightenment on your specific question see M. Mirzakhani's recent preprint on arXiv.org: 

