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