We investigate the possible homological classes of rational curves on
the moduli space $X_n=\overline{\mathcal{M}_{0,n}}$ of rational nodal
curves with $n$ marked points.
In the case of $X_5$ and $X_6$ the relevant homology classes belong to certain
polyhedral cones which we describe explicitly. Not all homology classes in the
cone correspond to rational curves, but all classes sufficiently far from
the walls of the cone do.
The proof for $X_5$ is explicit, the proof for $X_6$ was carried out using computers.
Joint work with Shachar Carmieli.
