I have heard that solar neutrino experiments have shown that neutrinos
oscillate between the three types. This supposedly proves that
neutrinos have mass. This seems to violate the conservation of mass,
or momentum, or both.

On the other hand if neutrinos were massless, they would quite
reasonably be able to oscillate without violating any conservation
laws (they can not change their spin direction, however; so there are
still two fundamental types of neutrinos.)

Why am I wrong?

Andrew Usher