Everyone knows it is close to $c$, but how close? What are the recent results?
Answer
Your question is equivalent to asking what the absolute mass of the neutrinos is, and the answer is currently unknown. We do have decent values for the differences of the squared masses for all three required neutrino states (one pair separated by about $7.7 \times 10^{-5}\text{ eV}^2$, and another one standing off from them by about $\pm 2.4 \times 10^{-3}\text{ eV}^2$ (but note the sign ambiguity a problem known as the mass hierarchy question)). This puts a lower bound on the mass of the most massive state at about $0.05\text{ eV}$, but puts no non-trivial bound on the lowest neutrino mass.
Supernova neutrinos may eventually be able to answer this by time-of-flight, but it depends on the theoretical understanding of exactly what is going on inside the exploding star.
There is some data from SN 1987a, but it is not of sufficient quality to provide even a rough answer.
The best we can say at this time is that they are known to be quite small, indeed. (And that is another problem to keep the theorists busy...)
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