  Motivated by several recent results on the geometry of the moduli spaces
$\bar{\Cal M}_{g,n}$ of stable curves of genus $g$ with $n$ marked points, here
we determine their birational structure for small values of $g$ and $n$ by
exploiting suitable plane models of the general curve. More precisely, we show
that ${\Cal M}_{g,n}$ is rational for $g=2$ and $1\le n\le 12$, $g=3$ and $1\le
n\le 14$, $g=4$ and $1\le n\le 15$, $g=5$ and $1\le n\le 12$.
