  In this article we explicitly determine the structure of the Weierstrass
semigroups $H(P)$ for any point $P$ of the Suzuki curve $\mathcal{S}_q$. As the
point $P$ varies, exactly two possibilities arise for $H(P)$: one for the
$\mathbb{F}_q$-rational points (already known in the literature), and one for
all remaining points. For this last case a minimal set of generators of $H(P)$
is also provided. As an application, we construct dual one-point codes from an
$\mathbb{F}_{q^4}\setminus\fq$-point whose parameters are better in some cases
than the ones constructed in a similar way from an $\fq$-rational point.
