If a neutrino has a rest frame, why can't a photon have a rest frame as well? Concerning Rest Frame Wikipedia states:  

For example, in the rest frame of a neutrino particle travelling from the Crab Nebula supernova to Earth the supernova occurred in the 11th Century AD only a short while before the light reached Earth, but in Earth's rest frame the event occurred about 6300 years earlier. 

If a neutrino has a rest frame, why can't a photon have one as well?
 A: Neutrinos have mass and travel slightly below light speed, therefore an inertial frame for the neutrino exists, while it doesn't exist for a massless photon which travels exactly at $c$. We don't know the masses of the neutrinos, but neutrino oscillations tell us that the three neutrino families must have a mass difference. For all we know, one of the three neutrinos could still be massless, but it is assumed that all neutrino families have a non-vanishing mass.
A: Photons travel at exactly light speed, rather than just close to light speed.  That means that for photons, the time and space axes collapse to a single axis, reducing the number of dimensions the photon "sees" to one.  Unfortunately, a single dimension is not sufficient for a frame of reference for four dimensional space time, so the photon cannot have a rest frame.
A: Disclaimer: in this answer “photon” refers to an excitation of electromagnetic field, not to a fundamental particle as understood in QED and Standard Model.
Possibly unexpectedly for some readers, a photon can have its rest frame in a homogeneous medium that moves w.r.t. the frame with speed equal to the light speed in this medium (that is less than c because of refraction). Note it does not cause the photon to become a massive particle, see What if we could give photons some mass? for details. These photons are not “true” particles obeying “standard” laws of dynamics at all.
