We used the Advanced Camera for Surveys onboard the Hubble Space Telescope to obtain high-resolution i-band images of the centres of 23 single galaxies, which were selected because they have Sloan Digital Sky Survey (SDSS) velocity dispersions larger than 350 km s−1. The surface brightness profiles of the most luminous of these objects (Mi < −24) have well-resolved ‘cores’ on scales of 150–1000 pc, and share similar properties to brightest cluster galaxies. The total luminosity of the galaxy is a better predictor of the core size than is the velocity dispersion. The correlations of luminosity and velocity dispersion with core size agree with those seen in previous studies of galaxy cores. Because of high velocity dispersions, our sample of galaxies can be expected to harbour the most massive black holes, and thus have large cores with large amounts of mass ejection. The mass deficits inferred from core-Sersic fits to the surface brightness profiles are approximately double the black hole masses inferred from the M•–σ relation and the same as those inferred from the M•–L relation. The less luminous galaxies (Mi > −23) tend to have steeper ‘power-law’ inner profiles, higher ellipticity, discier isophotes and bulge-to-total ratios of order 0.5 – all of which suggest that they are ‘fast rotators’ and rotational motions could have contaminated the velocity dispersion estimate. There are obvious dust features within about 300 pc of the centre in about 35 per cent of the sample, predominantly in power-law rather than core galaxies.