What's the significance of neutrino oscillations?
They say something very important about the nature of neutrinos. As per this related answer, the Standard Model doesn't really cover it. Then when you ask about it, people tend to talk about neutrino mass. What you don't much hear is why do neutrinos oscillate? IMHO the answer has to be that the neutrino is something like a travelling breather, see Wikipedia:
GFDL image by Georgiev DD, Papaioanou SN, Glazebrook JF. Neuronic system inside neurons: molecular biology and biophysics of neuronal microtubules. Biomedical Reviews 2004; 15: 67-75. DOI: 10.14748/bmr.v15.103a
Note that the neutrino isn't necessarily described by "a breather solution for the nonlinear Schrödinger equation", but IMHO this points the way. Also note that when you slow down a photon to below c, it exhibits an "effective mass". In similar vein if the neutrino oscillates such that its speed varies, its mass will vary. For an analogy, twang a washing-line to emulate a photon. Sight your eye down the washing line to see a transverse wave race away. To emulate a neutrino, twist the washing line with a pair of pliers, then let go.