We present Galaxy Evolution Explorer far-ultraviolet (FUV) and near-ultraviolet (NUV) imaging of 34 nearby early-type galaxies from the SAURON representative sample of 48 E/S0 galaxies, all of which have ground-based optical imaging from the MDM Observatory. The surface brightness profiles of nine galaxies (≈26 per cent) show regions with blue UV-optical colours suggesting RSF. Five of these (≈15 per cent) show blue integrated UV-optical colours that set them aside in the NUV integrated colour-magnitude relation. These are objects with either exceptionally intense and localized NUV fluxes or blue UV-optical colours throughout. They also have other properties confirming they have had RSF, in particular Hβ absorption higher than expected for a quiescent population and a higher CO detection rate. This suggests that residual star formation is more common in early-type galaxies than we are used to believe. NUV blue galaxies are generally drawn from the lower stellar velocity dispersion (σe < 200 km s-1) and thus lower dynamical mass part of the sample. We have also constructed the first UV Fundamental Planes and show that NUV blue galaxies bias the slopes and increase the scatters. If they are eliminated, the fits get closer to expectations from the virial theorem. Although our analysis is based on a limited sample, it seems that a dominant fraction of the tilt and scatter of the UV Fundamental Planes is due to the presence of young stars in preferentially low-mass early-type galaxies. Interestingly, the UV-optical radial colour profiles reveal a variety of behaviours, with many galaxies showing signs of RSF, a central UV-upturn phenomenon, smooth but large-scale age and metallicity gradients and in many cases a combination of these. In addition, FUV-NUV and FUV-V colours even bluer than those normally associated with UV-upturn galaxies are observed at the centre of some quiescent galaxies. Four out of the five UV-upturn galaxies are slow rotators. These objects should thus pose interesting challenges to stellar evolutionary models of the UV upturn.