In the quest for ultra-high-density magnetic recording, new materials in the nanometre range have attracted much interest over the last decade involving intense studies of L10 phases of contemporary or future storage media materials like FePt or CoPt nanoparticles. Based on large-scale density functional theory calculations, we provide a systematic overview of the structural and magnetic properties of various morphologies of FePt and CoPt nanoclusters with diameters up to 3 nm. In this size range, the ordered multiply twinned morphologies are energetically favoured over the nanoclusters with the desired layer type L10 and high magnetocrystalline anisotropy. Other nanoparticles of interest, like FePd, also show a preference for multiply twinned structures or exhibit, as in the case of MnPt nanoclusters, a strong tendency for antiferromagnetic ordering instead of ferromagnetic order. The compositional trends of the various nanoparticles can be traced back to differences in the partial electronic density of states of the 3d element.