  This paper is a contribution towards a solution for the longstanding open
problem of classifying linear systems of conics over finite fields initiated by
L. E. Dickson in 1908, through his study of the projective equivalence classes
of pencils of conics in $\mathrm{PG}(2,q)$, for $q$ odd. In this paper a set of
complete invariants is determined for the projective equivalence classes of
webs and of squabs of conics in $\mathrm{PG}(2,q)$, both for $q$ odd and even.
Our approach is mainly geometric, and involves a comprehensive study of the
geometric and combinatorial properties of the Veronese surface in
$\mathrm{PG}(5,q)$. The main contribution is the determination of the
distribution of the different types of hyperplanes incident with the $K$-orbit
representatives of points and lines of $\mathrm{PG}(5,q)$, where
$K\cong\mathrm{PGL}(3,q)$, is the subgroup of $\mathrm{PGL}(6,q)$ stabilizing
the Veronese surface.
