In a nutshell the state of the art measurement is that the difference between the second and third neutrino mass state is roughly 49.5 +/- 0.5 meV, and the difference between the first and second neutrino mass state is roughly 8.66 +/- 0.12 meV. The heaviest neutrino type cannot have a mass of less than 56 meV with 95% confidence, but since the hierarchy of the neutrino masses ("normal" or "inverted") and the absolute neutrino masses are not known, the exact neutrino masses are also not known.
A measurement comparing the time of arrival of light and neutrinos from a distant supernova in 1987 capped the maximum neutrino mass at 6 eV. Beta decay experiments, taken together with mass difference data from neutrino oscillation measurements, imply that none of the three neutrino masses can be more than 2 eV.
Planck satellite data combined with other astronomy data and included in a cosmology model imply that there are three neutrino types and that the sum of the mass of the three neutrino types cannot exceed 230 meV at 95% confidence. In a normal hierarchy, that would imply maximal neutrino masses of 57 meV, 66 meV and 115 meV. The best fit value for the sum of the three neutrino masses for this data is considerably smaller.
The minimum sum of the mass of the three neutrino types, at 95% confidence, from neutrino oscillation data is 64.4 meV. Using the best fit value for the three neutrino types, a "normal" hierarchy and a lowest neutrino mass of 1.2 meV, the sum of the three neutrino masses would be about 68 meV, which is very close to the best fit from the Planck and other astronomy data.
New experiments whose results will probably be available within the next five to ten years, and possibly sooner, will be able to tell us if the neutrino mass hierarchy is "normal" or "inverted", greatly reducing the uncertainty in the absolute neutrino mass measurement. A new supernova like the one in 1987 would also give us much more accurate information about neutrino mass than that one did because current astronomy observations are much more precise.
[A]1 MeV neutrino from a nuclear reaction with a mass of 1 eV travels at roughly (1-10-12) times the speed of light. For a 1 GeV neutrino from an accelerator this gets even worse with (1-10-18). While light travels 1000 kilometers, the neutrino just gets behind by 1 nanometer or 1 femtometer respectively.A previous OPERA report that neutrinos traveled slightly faster than the speed of light was ultimately determined to be due to a loose cable connection in their detector.
There is no way to see such a tiny difference, so measurements are expected to be consistent with the speed of light. Those measurement are still interesting: apart from light, the neutrinos should be the fastest things where we can measure the speed. It is a test of special relativity. In 2011 the OPERA collaboration announced that their measurements seem to indicate neutrinos that are faster than the speed of light by one part in 40,000. They later discovered that a loose cable spoiled clock synchronisation at one point, and published updated results that are in agreement with the speed of light. Due to the increased interest, multiple other experiments performed speed measurements, the most recent one (MINOS) this week. All results agree with the speed of light, and a deviation has to be smaller than 1 part in 500,000.
As noted above, since it is difficult to be sufficiently precise, Earth bound neutrino speed measurements are largely useless for purposes of determining neutrino mass.
But, these measurements do provide a good way to directly measure the constant "c" (the speed of light in a vacuum). And, they also provide excellent proof of Einstein's law of special relativity, because neutrinos whose momenta differ by a factor of thousand differ in speed by only a little less than one part per trillion, meaning that their speeds are indistinguishable to the most precise measurements of speed known to mankind.
A comment to the article notes that the T2K experiment's data is a best fit to a "normal" mass hierarchy and maximal CP violation, and references a conference presentation to that effect.
The experiment has seen 3 electron anti-neutrino events when 3.73 would be expected so far for a "normal" mass hierarchy and maximal CP violation (- pi/2), and more events would be expected in any other scenario. Specifically, in the normal hierarchy with no CP violation 4.32 events would be expected, and in a normal hierarchy with +pi/2 CP violation, 4.85 events would be expected. In an inverted hierarchy with maximal CP violation, 4.18 events would be expected, compared to 4.85 with no CP violation and 5.45 with +pi/2 CP violation.
Given that three of the possibilities round to 4 expected events, and that the other three round to 5 expected events, and that there is some random variation, an actual event count of 3 isn't terribly informative, but does tend to disfavor the scenarios that expect higher numbers of events so far. There is even a 20% change that all of the three measured events are actually only background noise, for which 0, 1 or 2 events would be much more likely, but 3 events wouldn't be that unusual. Even with more exactly analysis the odds that the results are just background noise can't be reduced below about 13%.
The Missing Pieces
The parameters of the simplest neutrino mixing model has three mixing angles, a CP violation phase, and the two mass eigenvalue differences described above, as well as a determination of whether the mass hierarchy is "normal" or "inverted" and the absolute masses of the neutrino types.
There are decent measurements of all three mixing angles and the two mass differences, but the "normal" v. "inverted" mass hierarchy, the absolute masses of the neutrion types, the CP violation phase, and the "quadrant" of one of the mixing angles (i.e. is it a few degrees over or under 45 degrees) remain unresolved.
Also unresolved is the question of whether the mass of neutrinos is "Dirac" like all of the other Standard Model fermions, or "Majorana" which would be the case if a neutrino was its own antiparticle. This best experimental data to resolve this is neutrinoless beta decay experiments which are not yet quite precise enough to resolve the issue. If neutrinos have Majorana mass (at least in part), then they can violate "lepton number" conservation in neutrinoless double beta decay events, and additional parameters not mentioned above are necessary to describe them.
Another unanswered question, which is important to cosmology, is the ratio of neutrinos to anti-neutrinos, and the ratio of neutrino generations, the universe. Since the number of neutrinos in the universe vastly outnumber the number of charged leptons in the universe, this ratio is critical to determining the aggregate lepton number of the universe (a quantity conserved in all interactions in the Standard Model except in vanishing rare high energy sphaleron interactions that are theoretically possible but have never been observed and still conserve the quantity B-L; they convert baryons to antileptons and antibaryons to leptons), which has important implications for cosmology and fundamental physics. But, while it is hard to measure a neutrino at all, it is harder still to determine if what one is measuring is a neutrino or an antineutrino, although the task is not in principle impossible, and there are hints that antineutrinos greatly outnumber neutrinos in the universe. In contrast, there are quite accurate estimates of the baryon number of the universe. It is not known if dark matter, if it exists, has either baryon number or lepton number or some equivalent conserved quantity (such as R-parity if dark matter consists of supersymmetric particles).
I predict that neutrinos will be found to have a "normal" mass hierarchy, with a lightest neutrino mass state of roughly 1 meV or less, that the quadrant of the mixing angle whose quadrant is undetermined will be a bit under 45 degrees rather than a bit over 45 degrees, that neutrinos will be found not to have Majorana mass, and that neutrino mixing will involve significant CP violation although probably not quite maximal CP violation.
While the answers may not be definitive by then, I strongly suspect that we will have strongly suggestive data on all of these questions by the year 2020.
Neutrinos and Sterile Neutrinos As Dark Matter
There has been some speculation that the universe also has "sterile neutrinos" which unlike normal neutrinos, do not interact via the weak force because they are "right handed" in the ordinary matter state, and "left handed" in the antimatter state, but which still might oscillate with ordinary neutrinos. I predict that no such particles exist, although it don't entirely rule out a "dark matter" particle of some mass that behaves like a sterile neutrino of some mass, but does not oscillate with the three ordinary neutrinos.
Ordinary neutrinos are not a fit to explain "dark matter" phenomena because they are too light, in the aggregate, to provide sufficient mass, and have too high an average speed due to their low mass, to give rise to the kind of dark matter phenomena driven structure observed in the universe. But, a single kind of keV mass scale particle with the properties of a sterile neutrino (except neutrino oscillation) would fit the experimental dark matter data fairly well.