It is a classical result that there are 12 (irreducible) rational cubic curves through 8 generic points in \(\mathbb {P}_{\mathbb {C}}^2\) , but little is known about the non-generic cases. The space of 8-point configurations is partitioned into strata depending on combinatorial objects we call quatroids, a higher-order version of representable matroids. We compute all 779,777 quatroids on eight distinct points in the plane, which produces a full description of the stratification. For each stratum, we generate several invariants, including the number of rational cubics through a generic configuration. As a byproduct of our investigation, we obtain a collection of results regarding the base loci of pencils of cubics and positive certificates for non-rationality.
