C60 monolayers grown on Ag(111) at room temperature form incommensurate lattices that convert into a commensurate (23 × 23)R30 phase upon annealing. The C60 molecules in the commensurate phase have been observed to exist in three different states on Ag(111), namely bright, dim, and superbright (SB). All three species are in dynamical equilibrium at 280 < T < 440 K. The bright and dim species were the subject of an earlier low-energy electron diffraction study that determined their geometries on the surface and the dynamics of the switching between those two states. The study presented here takes a closer look at the SB molecules, which appear and disappear at temperature-dependent rates with a measured activation barrier of 1.5 eV. The SB molecules in the commensurate phase comprise about 0.5% of the molecules and have a spatially random distribution. The evidence suggests that the formation of the three different states of C60 on Ag(111) is a result of stress imposed by the substrate as the C60 adopts the commensurate spacing that is slightly smaller than its natural spacing. In the incommensurate phases, there is no bright-dim contrast, but SB C60 molecules form and organize into ordered arrays that appear to correspond to the moiré patterns that are produced by the mutually incommensurate lattices. This suggests that the substrate responds to the nonuniform forces imposed by the C60 molecules by producing raised islands of Ag atoms at the vertices of the moiré structure. A similar island structure may account for the SB molecules in the commensurate phase. © 2014 American Physical Society.