  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 $779777$
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.
