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.