By a series of independent decisions, an international network of massive underground neutrino detectors is being established whose collective sensitivity to supernova neutrinos will be unprecedented. Individually and in coordinated fashion, these detectors could provide temporal, energetic, angular, and flavor information for any stellar collapse in our Galaxy, the Large Magellanic Cloud, or the Small Magellanic Cloud. In this paper, we construct a detailed fiducial model of supernova neutrino bursts that incorporates the numerous emission features derived and predicted by supernova and protoneutron star theorists during the last decade. With this generic model, we calculate and predict the response of representative \ifmmode \check{C}\else \v{C}\fi{}erenkov and scintillation neutrino telescopes to a nearby neutrino burst. The strengths and weaknesses of the various detectors are highlighted and explored. We focus on the information-rich features of the neutrino signal that were not detectable from SN 1987A. In addition, we investigate the effect of a nonzero neutrino mass on the detected signals. The latter study focuses on what can be derived with high-resolution observations of a galactic collapse.