  The atmospheric neutrino data collected by the IceCube experiment and its
low-energy extension DeepCore provide a unique opportunity to probe the
neutrino sector of the Standard Model. In the low energy range the experiment
have observed neutrino oscillations, and the high energy data are especially
sensitive to signatures of new physics in the neutrino sector. In this context,
we previously demonstrated the unmatched potential of the experiment to reveal
the existence of light sterile neutrinos. The studies are routinely performed
in the simplest 3+1 model concentrating on disappearance of muon neutrinos of
TeV energy as a result of their mixing with a sterile neutrino. We here extend
this analysis to include cascade events that are secondary electromagnetic and
hadronic showers produced by neutrinos of all flavors. We find that it is
possible to probe the complete parameter space of 3+1 model, including the
poorly constrained mixing of the sterile neutrino to tau neutrinos. We show
that $\nu_\tau-\nu_s$ mixing results into a unique signature in the data that
will allow IceCube to obtain constraints well below the current upper limits.
