Do you know a particle smaller than an electron neutrino? An electron neutrino is far by the smallest particle I know! Do you know something smaller than that?
The electron neutrino has the smallest non-zero mass of all known particles.
The electron neutrino doesn't have a well-defined mass. It's a superposition of the three neutrino mass eigenstates, the lightest of which is the lightest know fermion. However, it is not know whether the lightest neutrino has a non-zero mass.
If the lightest neutrino had zero mass, then the determinant of the mass matrix would be 0 and neutrino oscillations would not occur.
You are correct that the determinant would be 0; however, I believe that you are incorrect that that would eliminate oscillations. As far as I know, the only ways to prevent oscillations are to set mixing angles to 0 or to make neutrino mass states degenerate. Neither of these possibilities are guaranteed by a 0 determinant.