  We study the reciprocal position of nine points in the plane, according to
their collinearities. In particular, we consider the case in which the nine
points are contained in an irreducible cubic curve and we give their
classification. If we consider two configurations different when the associated
incidence structures are not isomorphic, we see that there are 131
configurations that can be realized in $\mathbb{P}^2_{\mathbb{Q}}$, and there
are two more in $\mathbb{P}^2_K$, where
  $K =\mathbb{Q}[\sqrt{-3}]$ (one of the two is the Hesse configuration given
by the nine inflection points of a cubic curve). Finally, we compute the
possible Hilbert functions of the ideals of the nine points.
