Let us briefly review the history of the discovery of neutrinos.
From Wikipedia: "Historically, the study of beta decay provided the first physical evidence of the neutrino. In 1911 Lise Meitner and Otto Hahn performed an experiment that showed that the energies of electrons emitted by beta decay had a continuous rather than discrete spectrum. This was in apparent contradiction to the law of conservation of energy, as it appeared that energy was lost in the beta decay process. A second problem was that the spin of the Nitrogen-14 atom was 1, in contradiction to the Rutherford prediction of ½.
In 1920-1927, Charles Drummond Ellis (along with James Chadwick and colleagues) established clearly that the beta decay spectrum is really continuous, ending all controversies.
In a famous letter written in 1930 Wolfgang Pauli suggested that in addition to electrons and protons atoms also contained an extremely light neutral particle which he called the neutron. He suggested that this "neutron" was also emitted during beta decay and had simply not yet been observed. In 1931 Enrico Fermi renamed Pauli's "neutron" to neutrino, and in 1934 Fermi published a very successful model of beta decay in which neutrinos were produced.
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http://en.wikipedia.org/wiki/Beta_decay
"Before the idea of neutrino oscillations came up, it was generally assumed that neutrinos travel at the speed of light. The question of neutrino velocity is closely related to their mass. According to relativity, if neutrinos are massless, they must travel at the speed of light. However, if they carry a mass, they cannot reach the speed of light.
http://en.wikipedia.org/wiki/Neutrino
In other words, in order to conserve both momentum and energy during beta decay, the theoretical particle called a 'neutrino' was predicted. It was presumed that the neutrino either travelled at the speed of light and had zero rest mass (most dominant theory until the 1980s) but momentum (analogous to the electromagnetic photon, speed of light, momentum, but zero rest mass); or else it travelled at near-relativistic speeds with very small rest-mass. (less popular and unproven).
However, with the apparent discovery of neutrino oscillation, it became popular though not universal to assert that neutrinos have a very small rest mass: "Neutrinos are most often created or detected with a well defined flavor (electron, muon, tau). However, in a phenomenon known as neutrino flavor oscillation, neutrinos are able to oscillate between the three available flavors while they propagate through space. Specifically, this occurs because the neutrino flavor eigenstates are not the same as the neutrino mass eigenstates (simply called 1, 2, 3). This allows for a neutrino that was produced as an electron neutrino at a given location to have a calculable probability to be detected as either a muon or tau neutrino after it has traveled to another location. This quantum mechanical effect was first hinted by the discrepancy between the number of electron neutrinos detected from the Sun's core failing to match the expected numbers, dubbed as the "solar neutrino problem". In the Standard Model the existence of flavor oscillations implies nonzero differences between the neutrino masses, because the amount of mixing between neutrino flavors at a given time depends on the differences in their squared-masses. There are other possibilities in which neutrino can oscillate even if they are massless. If Lorentz invariance is not an exact symmetry, neutrinos can experience Lorentz-violating oscillations."http://en.wikipedia.org/wiki/Neutrino
Thus, observed oscillations in 'flavor' (type of neutrino based on origin source) suggested that neutrinos had a small rest mass, and therefore according to Einstein had to travel at less than c. But do they?
"Lorentz-violating neutrino oscillation refers to the quantum phenomenon of neutrino oscillations described in a framework that allows the breakdown of Lorentz invariance. Today, neutrino oscillation or change of one type of neutrino into another is an experimentally verified fact; however, the details of the underlying theory responsible for these processes remain an open issue and an active field of study. The conventional model of neutrino oscillations assumes that neutrinos are massive, which provides a successful description of a wide variety of experiments; however, there are a few oscillation signals that cannot be accommodated within this model, which motivates the study of other descriptions. In a theory with Lorentz violation neutrinos can oscillate with and without masses and many other novel effects described below appear. The generalization of the theory by incorporating Lorentz violation has shown to provide alternative scenarios to explain all the established experimental data through the construction of global models."
http://en.wikipedia.org/wiki/Lorentz-violating_neutrino_oscillations
If they have a rest mass, and travel at near-c but slightly below c, there should be a slight variation in their speeds based upon their total energy (most of which would be kinetic energy, not rest-mass energy). In other words, various high-energy neutrinos would travel at, for example, .99999997 c or .99999995 c, etc., and this variation in speed, however slight, should be detectable.
So let us again turn to the 1987a supernova data.
http://en.wikipedia.org/wiki/Supernova_1987A
In this observation, a star core collapsed and released a lot of energy. Most of the excess energy is predicted in theory to be radiated away in a massive burst of neutrinos/anti-neutrinos formed from pair-production, and these neutrinos would be of all 3 flavors.
"The light from the supernova reached Earth on February 23, 1987." ... "Approximately three hours before the visible light from SN 1987A reached the Earth, a burst of neutrinos was observed at three separate neutrino observatories. This is likely due to neutrino emission (which occurs simultaneously with core collapse) preceding the emission of visible light (which occurs only after the shock wave reaches the stellar surface). At 7:35 a.m. Universal time, Kamiokande II detected 11 antineutrinos, IMB 8 antineutrinos and Baksan 5 antineutrinos, in a burst lasting less than 13 seconds."
In other words, these neutrinos travelled a total distance of 5.3 X 10^12 light seconds (168,000 light years), all originating at roughly the same time (within about a 10 second burst of neutrino emission), and all arrived at earth (the light-transit time of earth's diameter is less than 1 second, and is not a factor due to the spacing of the detectors) within about 10 seconds of each other. In other words, they all travelled at close to the same speed to within 12 orders of magnitude, far greater than any other measurement precision ever made for the speed of light. And, they all travelled at very close to the speed of light (travelling the same distance as the photons at a speed consistent to c to within about 1 part per 500 million).
One would expect that since the neutrinos are emitted with potentially a range of energies, that their transit time would have exhibited a range of speeds as mentioned at the start of this discussion. But that is not what was observed. The actual observation is consistent with neutrinos as having zero rest mass, and traveling at c, and inconsistent with having a rest-mass and ejected with a spectrum of varying energies.
It should be further noted, however, that "Approximately three hours earlier, the Mont Blanc liquid scintillator detected a five-neutrino burst, but this is generally not believed to be associated with SN 1987A.".
In other words, during the 1987a burst, 29 neutrinos were counted total (if the first five are counted as part of the burst); 1/6th (5) arrived 3 hours early (though not previously believed to be part of 1987a), and 5/6th (24) arrived at light speed (after taking into consideration the head-start they had over the photons). This might imply that 5 types of neutrinos travel at c, and 1 type travels at slightly above c.
This is all still very confusing, is it not?