Reduction type of non-hyperelliptic genus 3 curves

26 pages; Let $C/K: F=0$ be a smooth plane quartic over a complete discrete valuation field $K$ whose residue field is of characteristic different from $2,3, 5$ and $7$. We give various characterizations of the reduction (i.e. non-perelliptic genus 3 curve, hyperelliptic genus 3 curve or bad) of the stable model of $C$: in terms of the existence of a particular plane quartic model; in terms of the valuations of the theta constants of $C$; in terms of the valuations of the Dixmier-Ohno invariants of $C$. The last one leads to an easy computable criterion.