  Galactic halo neutralinos ($\chi$) captured by the Sun or Earth produce
high-energy neutrinos as end-products of various annihilation modes. These
neutrinos can travel from the Sun or Earth cores to the neighborhood of
underground detectors (``neutrino telescopes") where they can interact and
produce upwardly-moving muons. We compute these muon fluxes in the context of
the minimal $SU(5)$ supergravity model, and the no-scale and dilaton
$SU(5)\times U(1)$ 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 $SU(5)\times U(1)$ models is accessible for $m_\chi\sim m_{\rm Fe}$, which
in turn implies constraints for the lightest chargino mass around 100 GeV for a
range of $\tan\beta$ values. We also delineate the regions of parameter space
that would be accessible with the improvements of experimental sensitivity
expected in the near future at Gran Sasso, Super-Kamiokande, and other
facilities such as DUMAND and AMANDA, currently under construction. We conclude
that if neutralinos are present in the halo, then this technique can be used to
eventually explore more than half of the allowed parameter space of these
specific models, and more generally of a large class of supergravity models, in
many ways surpassing the reach of traditional collider experiments.
