The 3 neutrino families ($e$, $\mu$, $\tau$) are usually called neutrino flavors.
Neutrino flavor oscillation requires that the mass eigenstates of neutrinos are not equal and that the mass eigenstate is also not a flavor eigenstate. Since a neutrino is always produced in a flavor eigenstate (i.e. associated with an $e$, $\mu$, $\tau$), this flavor eigenstate wave function will be a mixture of the 3 mass eigenstates such that at the time of production it is a pure flavor eigenstate. However as the neutrino wave function propagates, the 3 mass eigenstates will effectively move at different speeds so that at the point in space where the propagating neutrino interacts with the measuring apparatus, it will be a different mixture of flavor eigenstates. Thus the possibility of flavor oscillation requires that the masses of the mass eigenstates are not equal.
That is why the flavor oscillation experiments always measure differences in masses (squared) but not absolute masses. So since oscillations among all three flavors have been observed, there must be 3 different mass eigenstates with different masses. However, the flavor oscillation experiments would allow the lightest mass eigenstate to have zero mass but would require that at least 2 mass eigenstates have masses that are non-zero and are not equal.
It is commonly assumed that all 3 masses are non-zero and not equal.
Note, for example, that you cannot talk about the mass of the electron neutrino since the flavor eigenstate of an electron neutrino will be a mixture of the 3 different mass eigenstates.