The simplest explanation for the atmospheric neutrino anomaly is the transition of muon neutrinos and antineutrinos into states of different flavor. Data from the reactor experiment Chooz, and the measurement of the energy and angle distribution of electron--like events in Super--Kamiokande put strong constraints on $��_��$--$��_e$ mixing, but the transitions $��_��\to ��_��$ and $��_��\to ��_s$ are viable possibilites. If the muon neutrinos are mixed with a singlet sterile state, the energy and zenith angle dependence of the oscillation probability is modified by matter effects, that are important for $E_��/|��m^2| \aprge 10^3$ GeV/eV$^2$. We confront the theoretical predictions with existing data on contained events and neutrino--induced upward going muons.

32 pages, Latex with included 16 figures