Context. Our knowledge of short gamma-ray bursts (GRBs) has significatively improved in the Swift era. Rapid multiband observations from the largest ground-based observatories led to the discovery of the optical afterglows and host galaxies of these events. In spite of these advancements, the number of short GRBs with secure detections in the optical is still fairly small. Short GRBs are commonly thought to originate from the merging of double compact object binaries but direct evidence for this scenario is still missing. Aims. Optical observations of short GRBs allow us to measure redshifts, firmly identify host galaxies, characterize their properties, and accurately localize GRBs within them. Multiwavelength observations of GRB afterglows provide useful information on the emission mechanisms at work. These are all key issues that allow one to discriminate among different models of these elusive events. Methods. We carried out photometric observations of the short/hard GRB 051227, GRB 061006, and GRB 071227 with the ESO-VLT starting from several hours after the explosion down to the host galaxy level several days later. For GRB 061006 and GRB 071227 we also obtained spectroscopic observations of the host galaxy. We compared the results obtained from our optical observations with the available X-ray data of these bursts. Results. For all the three above bursts, we discovered optical afterglows and firmly identified their host galaxies. About half a day after the burst, the optical afterglows of GRB 051227 and GRB 061006 present a decay significatly steeper than in the X-rays. In the case of GRB 051227, the optical decay is so steep that it likely indicates different emission mechanisms in the two wavelength ranges. The three hosts are blue star forming galaxies at moderate redshifts and with metallicities comparable to the Solar one. The projected offsets of the optical afterglows from their host galaxy center span a wide range, but all afterglows lie within the light of their hosts and present evidence for local absorption in their X-ray spectra. We discuss our findings in light of the current models of short GRB progenitors.