Why does this happen for neutrinos and not for say, electrons and muons. Is there some way to predict which particles might oscillate amongst their flavour and which won't?
To make it short, you need "proper" Mass/Energy Eigenstates and "proper" flavour Eigenstates.
The mixing theory means that you have real coherent states. This works very well since the mass differences of Neutrinos are extremely small and the Energy is high.
For the charged leptons this is not the case. For an electron to turn into a Tau it needs a lot of energy. Consider a beam of e- with an energy just above the $\tau$- rest mass and along its path some of the e- turn into $\tau$- The other e- would quickly overtake and the distance between the particles would be huge. This makes it difficult to get a coherent state.
In the ultrarelativistic limit this could work in principle. But has not been observed. AFAIK the mixing angle is a function of the mass difference and thus, it is extremely small for charged leptons.
Because of all this, we just pretend that the charged lepton mass Eigenstates are equal to the flavour Eigenstates and we thus give those three masses the Name "Electron", "Muon" and "Tau"