We present an analysis of 13 of the best quality ultraluminous X-ray source (ULX) data sets available from XMM-Newton European Photon Imaging Camera (EPIC) observations. We utilize the high signal-to-noise in these ULX spectra to investigate the best descriptions of their spectral shape in the 0.3-10 keV range. Simple models of an absorbed power law or multicolour disc blackbody prove inadequate at describing the spectra. Better fits are found using a combination of these two components, with both variants of this model - a cool (~0.2 keV) disc blackbody plus hard power-law continuum, and a soft power-law continuum, dominant at low energies, plus a warm (~1.7 keV) disc blackbody - providing good fits to 8/13 ULX spectra. However, by examining the data above 2 keV, we find evidence for curvature in the majority of data sets (8/13 with at least marginal detections), inconsistent with the dominance of a power law in this regime. In fact, the most successful empirical description of the spectra proved to be a combination of a cool (~0.2 keV) classic blackbody spectrum, plus a warm disc blackbody that fits acceptably to 10/13 ULXs. The best overall fits are provided by a physically self-consistent accretion disc plus Comptonized corona model (DISKPN + EQPAIR), which fits acceptably to 11/13 ULXs. This model provides a physical explanation for the spectral curvature, namely that it originates in an optically thick corona, though the accretion disc photons seeding this corona still originate in an apparently cool disc. We note similarities between this fit and models of Galactic black hole binaries at high accretion rates, most notably the model of Done & Kubota. In this scenario the inner disc and corona become energetically coupled at high accretion rates, resulting in a cooled accretion disc and optically thick corona. We conclude that this analysis of the best spectral data for ULXs shows it to be plausible that the majority of the population are high accretion rate stellar-mass (perhaps up to 80 solar mass) black holes, though we cannot categorically rule out the presence of larger, ~1000 solar mass intermediate-mass black holes (IMBHs) in individual sources with the current X-ray data.