  Using a data sample of 980 fb$^{-1}$ collected with the Belle detector at the
KEKB asymmetric-energy $e^+e^-$ collider, we study for the first time the
singly Cabibbo-suppressed decays $\Omega^0_c\to\Xi^{-}\pi^{+}$ and
$\Omega^-K^+$ and the doubly Cabibbo-suppressed decay $\Omega^0_c\to \Xi^-
K^{+}$. Evidence for an $\Omega^0_c$ signal in the $\Omega^0_c\to\Xi^-\pi^+$
mode is reported with a significance of $4.5\sigma$ including systematic
uncertainties. The ratio of branching fractions to the normalization mode
$\Omega_c^0\to \Omega^-\pi^+$ is measured to be $${\cal B}(\Omega_{c}^{0} \to
\Xi^{-} \pi ^{+} )/{\cal B}(\Omega_c^0\to \Omega^-\pi^+)=0.253\pm 0.053({\rm
stat.})\pm 0.030({\rm syst.}).$$ No significant signals of
$\Omega^0_c\to\Xi^-K^+$ and $\Omega^-K^+$ modes are found. The upper limits at
$90\%$ confidence level on ratios of branching fractions are determined to be
$${\cal B}(\Omega_{c}^{0} \to \Xi^{-} K ^{+})/{\cal B}(\Omega_c^0\to
\Omega^-\pi^+) < 0.070$$ and $${\cal B}(\Omega_{c}^{0} \to
\Omega^{-}K^{+})/{\cal B}(\Omega_c^0\to \Omega^-\pi^+) < 0.29.$$
