Galactic halo neutralinos (chi) captured by the Sun or Earth produce h
igh-energy neutrinos as end products of various annihilation modes. Th
ese neutrinos can travel from the Sun or Earth cores to the neighborho
od of underground detectors (''neutrino telescopes'') where they can i
nteract and produce upwardly moving muons. We compute these muon fluxe
s in the context of the minimal SU(5) supergravity model, and the no-s
cale and dilaton SU(5)xU(1) supergravity models. At present, with the
Kamiokande 90% C.L. upper limits on the flux, only a small fraction of
the parameter space of the SU(5)xU(1) models is accessible for m(chi)
approximately m(Fe), which in turn implies constraints for the lighte
st chargino mass around 100 GeV for a range of tanbeta values. We also
delineate the regions of parameter space that would be accessible wit
h the improvements of experimental sensitivity expected in the near fu
ture at Gran Sasso, Super-Kamiokande, and other facilities such as DUM
AND and AMANDA, currently under construction. We conclude that if neut
ralinos are present in the halo, then this technique can be used to ev
entually explore more than half of the allowed parameter space of thes
e specific models, and more generally of a large class of supergravity
models, in many ways surpassing the reach of traditional collider exp
eriments.