We have used high-resolution, Hubble Space Telescope, near-infrared imaging to conduct a detailed analysis of the morphological properties of the most massive galaxies at high redshift, modelling the WFC3/IR H160-band images of the ≃200 galaxies in the CANDELS-UDS field with photometric redshifts 1 < z < 3, and stellar masses M* > 1011 M⊙. We have explored the results of fitting single-Sérsic and bulge+disc models, and have investigated the additional errors and potential biases introduced by uncertainties in the background and the on-image point spread function. This approach has enabled us to obtain formally acceptable model fits to the WFC3/IR images of >90 per cent of the galaxies. Our results indicate that these massive galaxies at 1 < z < 3 lie both on and below the local size–mass relation, with a median effective radius of ∼2.6 kpc, a factor of ≃2.3 smaller than comparably massive local galaxies. Moreover, we find that bulge-dominated objects in particular show evidence for a growing bimodality in the size–mass relation with increasing redshift, and by z > 2 the compact bulges display effective radii a factor of ≃4 smaller than local ellipticals of comparable mass. These trends also appear to extend to the bulge components of disc-dominated galaxies. In addition, we find that, while such massive galaxies at low redshift are generally bulge-dominated, at redshifts 1 < z < 2 they are predominantly mixed bulge+disc systems, and by z > 2 they are mostly disc-dominated. The majority of the disc-dominated galaxies are actively forming stars, although this is also true for many of the bulge-dominated systems. Interestingly, however, while most of the quiescent galaxies are bulge-dominated, we find that a significant fraction (25–40 per cent) of the most quiescent galaxies, with specific star formation rates sSFR < 10−10 yr−1, have disc-dominated morphologies. Thus, while our results show that the massive galaxy population is undergoing dramatic changes at this crucial epoch, they also suggest that the physical mechanisms which quench star formation activity are not simply connected to those responsible for the morphological transformation of massive galaxies into present-day giant ellipticals.