> Is there a set of neutrino masses that is minimally-illuminating of physics beyond the standard model?
There are no neutrino masses at all in the Standard Model. You can put them in by hand, but then you break half the nice ideas used to construct the Standard Model in the first place. Or you can put them in more properly, but then there are perhaps hundreds of distinct ways to do it, and we don't know which one is realized in nature. In other words, the state we're at with neutrinos is roughly where we were at with everything else in the 1960s. In some sense neutrinos have been "new physics" for decades -- we didn't understand them 30 years ago and we don't now.
As for specific mass measurements, there are theories that predict exceptionally nice values for the neutrino mass, but also plenty that don't have any such order. It's certainly possible that knowing all the neutrino masses to high precision wouldn't help at all. After all, we know the quark masses to high precision, and that precision hasn't helped us find an origin for the pattern in their masses. We just have to take them as free parameters.
There are no neutrino masses at all in the Standard Model. You can put them in by hand, but then you break half the nice ideas used to construct the Standard Model in the first place. Or you can put them in more properly, but then there are perhaps hundreds of distinct ways to do it, and we don't know which one is realized in nature. In other words, the state we're at with neutrinos is roughly where we were at with everything else in the 1960s. In some sense neutrinos have been "new physics" for decades -- we didn't understand them 30 years ago and we don't now.
As for specific mass measurements, there are theories that predict exceptionally nice values for the neutrino mass, but also plenty that don't have any such order. It's certainly possible that knowing all the neutrino masses to high precision wouldn't help at all. After all, we know the quark masses to high precision, and that precision hasn't helped us find an origin for the pattern in their masses. We just have to take them as free parameters.