Density-functional theory calculations have been carried out to study the adsorption of methanethiolate and atomic sulfur as a nonmolecular reference at the Cu(111) surface. A large number of surface models have been investigated considering a variety of binding sites and coverages at the ideal and reconstructed surface. For methanethiolate, we find that the proposed [(5)(0)(1)(3)] supercell commonly used to approximate the experimentally observed noncommensurate pseudo(100) reconstruction yields the lowest surface energy, but several similar local minima exist differing in the positions of the copper atoms. None of these structures show the regular nearly square coordination of the thiolate species observed in scanning tunneling microscopy (STM). Modifying the chemical composition of the relaxed layer, e.g., by adding another copper atom, yields structures of comparable stability. It is thus very likely that the proposed supercell is not a good approximation to the true pseudo(100) phase and that larger unit cells are needed to allow for a realistic relaxation of the reconstructed layer. For atomic sulfur, it is well established that the most stable phase at Cu(111) is a (root 7 X root 7)R19.1 degrees reconstruction. Its structure, however, has been discussed controversially in the literature for many years. While there is a consensus that the unit cell contains three sulfur atoms, there are still several competing models differing in the number of copper adatoms in the reconstructed layer. We find that three models have a very similar stability, and a three-copper adatom model is only marginally preferred. These results will be of importance for many fields from heterogeneous catalysis to covalent mechanochemistry and molecular nanomechanics.