We discuss the multiple technical choices that have to be made in ab initio density-functional calculations of the magnetic anisotropy of supported nanostructures: (i) choice of the exchange-correlation functional, (ii) degree of optimization of the geometry of the adsorbate/substrate complex, (iii) magnetic anisotropy energy calculated self-consistently or via the 'force theorem', (iv) calculations based on slab models of the substrate or using a Green's function describing a semi-infinite substrate, (v) full potential approach or atomic-sphere approximation. Using isolated Fe and Co atoms on Pt(111) as an example we demonstrate that by using a judicious combination of relatively crude approximations (complete neglect of structural relaxation, local exchange-correlation functional,...) seemingly good agreement with experimental anisotropy energies can be achieved, while the calculated orbital moments remain small. At a higher level of theory (relaxed adsorbate/substrate complex, gradient-corrected functionals,...) providing a realistic geometry of the adsorbate/substrate complex and hence a correct description of the interaction between the magnetic adatom and its ligands, anisotropy energies are also in semi-quantitative agreement with experiment, while the orbital moments of the adatoms are much too small. We suggest that the anisotropy energies provided by both approaches should be considered as lower limits of the real anisotropies. Without relaxation the ligand effect coupling the orbital moments of the adatom to the heavy atoms of the substrate is underestimated, while in a relaxed adsorbate/substrate complex the lack of orbital dependence of the exchange potential combined with a strong hybridization of adatom and substrate states leads to a strong underestimation of the orbital moment. We have briefly explored the influence of post-density-functional corrections. Adding a modest on-site Coulomb repulsion to the d states of the adatom (in a DFT+U approach) leads to a modest increase of spin and orbital moments of the adatom accompanied by a slow decrease of the induced moments, leaving the anisotropy energy almost unchanged.