It has been recently proposed that a sterile neutrino ν_h of mass m_h=40--80 MeV, mixing |U_{μh}|^2=10^{-3}--10^{-2}, lifetime τ_h \lsim 10^{-9} s, and a dominant decay mode (ν_h \to ν_μγ) could be the origin of the experimental anomalies observed at LSND, KARMEN and MiniBooNE. Such a particle would be abundant inside air showers, as it can be produced in kaon decays (K -> ν_h μ, K_L -> ν_h πμ). We use the Z-moment method to evaluate its atmospheric flux and the frequency of its decays inside neutrino telescopes. We show that the ν_h would imply around 10^4 contained showers per year inside a 0.03 km^3 telescope like ANTARES or the DeepCore in IceCube. These events would have a characteristic energy and zenith-angle distribution (E_ν= 0.1--10 TeV and θ< 90^o), which results from a balance between the reach of the heavy neutrino (that disfavors low energies) and a sizeable production rate and decay probability. The standard background from contained neutrino events (ν_e N \to e X and neutral-current interactions of high inelasticity) is 100 times smaller. Therefore, although it may be challenging from an experimental point of view, a search at ANTARES and IceCube could confirm this heavy-neutrino possibility.

10 pages. Comments on constraints from muon capture and cosmology added, minor corrections, references added. Version to appear as a Rapid Communication in PRD