It has been hypothesized that large fluxes of neutrinos may be created in astrophysical "cosmic accelerators." The primary background for a search for astrophysical neutrinos comes from atmospheric neutrinos, which do not exhibit the pointlike directional clustering that characterizes a distant astrophysical signal. We perform a search for neutrino point sources using the upward-going muon data from three phases of operation (SK-I, SK-II, and SK-III) spanning 2623 days of live time taken from April 1, 1996 to August 11, 2007. The search looks for signals from suspected galactic and extragalactic sources, transient sources, and unexpected sources. We find interesting signatures from two objects--RX J1713.7-3946 (97.5% CL) and GRB 991004D (95.3% CL)--but the signatures lack compelling statistical significance given trial factors. We set limits on the flux and fluence of neutrino point sources above energies of 1.6 GeV.