If a neutrino has mass then it travels less than the speed of light. Suppose I boost myself to the rest frame; i.e. bring it to rest in the laboratory. Now if it oscillates between different states and masses sitting there, where does the oscillating excess/loss energy and mass reside? In some internal state? I have a very limited knowledge of QM but in SR it would seem strange that "mass" would move into another place. A pointer to where I should start studying would (I hope) be sufficient.
OK, let's accept your conceit.
You ran after $\nu_1$, the lightest neutrino, for the sake of argument, and jumped on it. You ran real fast, as your Lorentz factor γ is several millions, (In natural units $\hbar=c=1$, in the lab frame, $E=p+m_1^2/(2p)+...$, not yours). You watch what is happening around you. You do not oscillate to anybody: You are a non-interacting plane wave. You will not change your energy or momentum--how could you? Near you, there are two other plane waves/neutrinos that you started your trip with, with different masses, energies and momenta, but only by a little!
They are related to you, as they were produced by the same pion decay, in your parent superposition $\nu_\mu$, a fictitious convenience superposition state helping with the accounting. Since they have slightly different masses, $m_2, m_3$, they are zipping by at slightly different energies and momenta, which they took from the pion at the moment of their birth, (e.g. as the e or μ in a K decay take different energies and momenta away, but since their masses are so enormously different, one would not bother superposing them!) Now there are sticklers who would arrange some uncertainty in the time you got emitted, or the position your parent decay occurred, to artificially arrange for a common momentum, or energy, or speed with your 2 fellow travelers. But you are going so fast, it hardly matters, to this order of the calculation.
The crucial thing, however, is that you and your two partners' waves will evince "beats" about the common average frequency and momentum, without any interaction, beyond the mere superposition--Fourier analysis: the envelope of the average wavetrain will show some oscillation with the distance of your trip. That is differences in your formation with your two partners: you'll just keep your steady course, and not exchange any momentum or energy or anything else.
And then you get detected: you deposit your energy and momentum to the observing atom. And so do your two partners. Nobody observed you during your trip, so you were virtual (!) but, still, since the trip was so long and the time of it likewise (yes! in our units) you were all but on-shell, close to the energy-momentum relation dictated by your mass. The shape of the group wave envelope, so the altered relationships with your partners will dictate the type of fictitious interaction combination ("flavor eigenstate") likely to hit the detector, but, in an oscillation experiment the parameters are chosen so it may be different than your birth flavor eigenstate. The atom that was hit, did not know what hit it, even though it would get a different energy and momentum from each of you three (by a little... so little it is hardly accounted for...) it registers being hit by a given flavor eigenstate formation of p~E, but it does not make much sense to ascribe a mass to that formation; only the component mass eigenstates, you and your 2 partners, had a mass.
The effect is kinematic, linear Fourier analysis beats, very similar to the musical phenomenon, except now it is relativistic.
The straightforward math for all this in in the PDG summary, eqns (14.9-15).
When you say "Suppose I boost myself to the rest frame" you have in effect asserted that you are looking at a particular mass state rather than at the superposition of mass-states that arises from allowing a flavor state to propagate freely.
This is related to the question of how neutrinos can oscillate though the lepton flavors have differing masses (also examined in "Neutrino Oscillations and Conservation of Momentum").
Boris Kayser's explanation of the latter problem has the flavor of consistent histories to me, though I suppose that you can also understand it in therms of the uncertainty principle.
As an experimental limitation, say that you have achieved the rest frame of a neutrino. How will you know? You'd have to measure it's interaction with something, but the cross-sections are vanishing even by the understanding of neutrino physics and the energies involved are likely much less than 1 eV. It is very challenging even in principle.