There are several motivations to search for possible non-zero neutrino masses [1].
Fermion masses in general are one of the major mysteries/problems of the standard model. Observation or nonobservation of the (oddball) neutrino masses could introduce a useful new perspective on the subject.
Nonzero masses are predicted in most extensions of the standard model. They therefore constitute a powerful probe of new physics.
There may be a hot dark matter component to the universe. If so, neutrinos would be (one of) the most important things in the universe.
The observed spectral distortion and deficit of solar neutrinos is most easily accounted for by the oscillations/conversions of a massive neutrino.
The ratio of atmospheric may be suggestive of neutrino oscillations.
There are hints for possible oscillations from the LSND experiment at LAMPF.
With or without neutrino mass and oscillations, the solar neutrino flux is (with helioseismology) one of the two known probes of the solar core. A similar statement applies to Type-II supernovae.
Although there are strong motivations for neutrino mass and mixing from theory, cosmology, and astrophysics, the number of types of neutrinos is limited. The LEP lineshape measurements imply that there are only three ordinary light neutrinos, and big bang nucleosynthesis severely constrains the parameters of possible sterile neutrinos (which interact and are produced only by mixing). There are only a limited range of well-motivated possibilities for neutrino masses and mixings. The new generations of laboratory and solar neutrino experiments should be able to cover this range and either clearly establish non-zero masses (probably the first break with the standard model) or else falsify the interesting possibilities.