  We analyze the possibility of superluminal neutrino propagation delta v = (v
- c)/c > 0 as indicated by OPERA data, in view of previous phenomenological
constraints from supernova SN1987a and gravitational Cerenkov radiation. We
argue that the SN1987a data rule out delta v ~ (E_\nu/M_N)^N for N \le 2 and
exclude, in particular, a Lorentz-invariant interpretation in terms of a
'conventional' tachyonic neutrino. We present two toy Lorentz-violating
theoretical models, one a Lifshitz-type fermion model with superluminality
depending quadratically on energy, and the other a Lorentz-violating
modification of a massless Abelian gauge theory with axial-vector couplings to
fermions. In the presence of an appropriate background field, fermions may
propagate superluminally or subluminally, depending inversely on energy, and on
direction. Reconciling OPERA with SN1987a would require this background field
to depend on location.
