The search for sterile neutrinos is motivated by the LSND and MiniBooNE, reactor and Gallium anomalies. The fact that this evidence is partly conflicting can be a consequence of either experimental systematics or of non-standard neutrino properties. A particular interesting case for such non-standard properties are scenarios where the simple dispersion relation between energy and momentum is changed due to the presence of additional terms which can result from Lorentz violation, shortcuts in extra dimension, the variation of mass, or non-standard matter effects. The consequences for neutrino oscillation phenomenology and cosmology are discussed.