  The number $N_9(5)$, the maximal number of $\mathbb{F}_9$-rational points on
curves over $\mathbb{F}_9$ of genus $5$ is unknown, but it is known that $32
\le N_9(5)\le 35$. In this paper, we enumerate hyperelliptic curves and
trigonal curves over $\mathbb{F}_3$ which have many $\mathbb{F}_9$-rational
points (and $\mathbb{F}_3$-rational points), especially the maximal number of
$\mathbb{F}_9$-rational points of those curves is $30$. Kudo-Harashita studied
the nonhyperelliptic and nontrigonal case,where they found a new example of
curves (over $\mathbb{F}_3$) of genus five which attains $32$ and proved that
there is no example attaining more than $32$, among sextic plane curves with
mild singularities. We conclude from the main results in this paper that we
need to search sextic models (i.e., nonhyperelliptic and nontrigonal) with bad
singularities, in order to find a genus-five curve over $\mathbb{F}_3$ with at
least $33$ $\mathbb{F}_9$-rational points.
