  Let $K$ be a local field and $f(x)\in K[x]$ be a non-constant polynomial. The
local zeta function $Z_f(s, \chi)$ was first introduced by Weil, then studied
in detail by Igusa. When ${\rm char}(K)=0$, Igusa proved that $Z_f(s, \chi)$ is
a rational function of $q^{-s}$ by using the resolution of singularities. Later
on, Denef gave another proof of this remarkable result. However, if ${\rm
char}(K)>0$, the question of rationality of $Z_f(s, \chi)$ is still kept open.
Actually, there are only a few known results so far. In this paper, we
investigate the local zeta functions of two-variable polynomial $g(x, y)$,
where $g(x, y)=0$ is the superelliptic curve with coefficients in a
non-archimedean local field of positive characteristic. By using the notable
Igusa's stationary phase formula and with the help of some results due to Denef
and Z${\rm \acute{u}}$${\rm\tilde{n}}$iga-Galindo, and developing a detailed
analysis, we prove the rationality of these local zeta functions and also
describe explicitly all their candidate poles.
