  We study genus 2 curves over finite fields of small characteristic. The
$p$-rank $f$ of a curve induces a stratification of the coarse moduli space
$\mathcal{M}_2$ of genus 2 curves up to isomorphism. We are interested in the
size of those strata for all $f \in \{0,1,2\}$. In characteristic 2 and 3,
previous results show that the supersingular $f=0$ stratum has size $q$. We
show that for $q=3^r$, over $\mathbb{F}_q$ the non-ordinary $f=1$ and ordinary
$f=2$ strata are of size $q(q-1)$ and $q^2(q-1)$, respectively. We give results
found from computer calculations which suggest that these formulas hold for all
$p \leq 7$ and break down for $p > 7$.
