  In this article we consider the moduli space of smooth $n$-pointed
non-hyperelliptic curves of genus 3. In the pursuit of cohomological
information about this space, we make $\mathbb{S}_n$-equivariant counts of its
numbers of points defined over finite fields for $n \leq 7$. Combining this
with results on the moduli spaces of smooth pointed curves of genus 0, 1 and 2,
and the moduli space of smooth hyperelliptic curves of genus 3, we can
determine the $\mathbb{S}_n$-equivariant Galois and Hodge structure of the
($\ell$-adic respectively Betti) cohomology of the moduli space of stable
curves of genus 3 for $n \leq 5$ (to obtain $n \leq 7$ we would need counts of
``8-pointed curves of genus 2'').
