We present observations of the afterglows and host galaxies of three short-duration gamma-ray bursts (GRBs): 100625A, 101219A, and 110112A. We find that GRB?100625A occurred in a z = 0.452 early-type galaxy with a stellar mass of ≈4.6 × 109 M ☉ and a stellar population age of ≈0.7 Gyr, and GRB?101219A originated in a star-forming galaxy at z = 0.718 with a stellar mass of ≈1.4 × 109 M ☉, a star formation rate of ≈16 M ☉?yr–1, and a stellar population age of ≈50?Myr. We also report the discovery of the optical afterglow of GRB 110112A, which lacks a coincident host galaxy to i 26?mag, and we cannot conclusively identify any field galaxy as a possible host. From afterglow modeling, the bursts have inferred circumburst densities of ≈10–4-1?cm–3 and isotropic-equivalent gamma-ray and kinetic energies of ≈1050-1051?erg. These three events highlight the diversity of galactic environments that host short GRBs. To quantify this diversity, we use the sample of 36 Swift short GRBs with robust associations to an environment (~1/2 of 68 short bursts detected by Swift to 2012 May) and classify bursts originating from four types of environments: late-type (≈50%), early-type (≈15%), inconclusive (≈20%), and "host-less" (lacking a coincident host galaxy to limits of 26?mag; ≈15%). To find likely ranges for the true late- and early-type fractions, we assign each of the host-less bursts to either the late- or early-type category using probabilistic arguments and consider the scenario that all hosts in the inconclusive category are early-type galaxies to set an upper bound on the early-type fraction. We calculate most likely ranges for the late- and early-type fractions of ≈60%-80% and ≈20%-40%, respectively. We find no clear trend between gamma-ray duration and host type. We also find no change to the fractions when excluding events recently claimed as possible contaminants from the long GRB/collapsar population. Our reported demographics are consistent with a short GRB rate driven by both stellar mass and star formation.
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