  Neutrino telescopes provide strong sensitivity to sterile neutrino
oscillations through matter-enhanced oscillation, occurring in the few TeV
energy range for eV$^{2}$-scale neutrino mass-squared splittings. Prior
searches have focused on $\nu_\mu$ disappearance, which has a particularly
strong sensitivity to the mixing angle $\theta_{24}$ via
$\nu_\mu\rightarrow\nu_s$ transitions. Nowadays, the $\nu_\mu\rightarrow\nu_e$
and $\nu_\mu\rightarrow\nu_\tau$ appearance channels have been considered less
promising at neutrino telescopes, due in part to the much smaller target volume
for cascades relative to tracks. This work explores the detectability of these
signatures at neutrino telescopes given present constraints on sterile neutrino
mixing, and as an example, forecasts the sensitivity of the IceCube Neutrino
Observatory to the mixing angles $\theta_{14}$, $\theta_{24}$, and
$\theta_{34}$ in the 3+1 sterile neutrino model using the cascade channel with
ten years of data. We find that $\nu_\tau$ appearance signatures consistent
with the existing IceCube $\nu_\mu$ disappearance best-fit point are
discoverable for values of $\theta_{34}$ consistent with world constraints, and
that the sterile neutrino parameters favored by the BEST and gallium anomalies
are expected to be testable at the 95\% confidence level.
