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Precise tests of Lorentz invariance in neutrinos can be performed using longbaseline experiments such as MINOS and OPERA or neutrinos from astrophysicalsources. The MINOS collaboration reported a measurement of the muonic neutrinovelocities that hints to super-luminal propagation, very recently confirmed at6 sigma by OPERA. We consider a general parametrisation which goes beyond theusual linear or quadratic violation considered in quantum-gravitational models.We also propose a toy model showing why Lorentz violation can be specific tothe neutrino sector and give rise to a generic energy behaviour E^alpha, wherealpha is not necessarily an integer number. Supernova bounds and the preferredMINOS and OPERA regions show a tension, due to the absence of shape distortionin the neutrino bunch in the far detector of MINOS. The energy independence ofthe effect has also been pointed out by the OPERA results.

Superluminal neutrinos in long baseline experiments and SN1987a