We present an extensive study of accretion onto neutron stars in which the velocity of the neutron star and structure of the surrounding medium is such that the Bondi-Hoyle accretion exceeds 10(-4) M. yr(-1). Two types of initial conditions are considered for a range of entropies and chemical compositions: an atmosphere in pressure equilibrium above the neutron star, and a freely falling inflow of matter from infinity (also parameterized by the infall rate). We then evolve the system with one- and two-dimensional hydrodynamic codes to determine the outcome. For most cases, hypercritical (also termed ''super Eddington'') accretion caused by rapid neutrino cooling allows the neutron star to accrete above the Bondi-Hoyle rate as previously pointed out by Chevalier. However, for a subset of simulations which corresponds to evolutionarily common events, convection driven by neutrino heating can lead to explosions by a mechanism similar to that found in core-collapse supernovae. Armed with the results from our calculations, we are in a position to predict the fate of a range of rapid-infall neutron star accretors present in certain low-mass X-ray binaries, common envelope systems, supernova fallbacks, and Thorne-Zytkow objects (TZOs). A majority of the common envelope systems that we considered led to explosions expelling the envelope, halting the neutron star's inward spiral, and allowing the formation of close binary systems. As a result, the smothered neutron stars produced in the collisions studied by Davies & Benz may also explode, probably preventing them from forming millisecond pulsars. For the most massive supernovae, in which the fallback of material toward the neutron star after a successful explosion is large, we find that a black hole is formed in a few seconds. Finally, we argue that the current set of TZO formation scenarios is inadequate and leads instead to hypercritical accretion and black hole formation. Moreover, it appears that many of the current TZ models have structures ill suited for modeling by mixing-length convection. This has prompted us to develop a simple test to determine the viability of this approximation for a variety of convective systems.
