The relative energies of side-on versus end-on binding of molecular oxygen to a supported Cu(I) species, and the singlet versus triplet nature of the ground electronic state, are sensitive to the nature of the supporting ligands and, in particular, depend upon their geometric arrangement relative to the O-2 binding site. Highly correlated ab initio and density functional theory electronic structure calculations demonstrate that optimal overlap (and oxidative charge transfer) occurs for the side-on geometry, and this is promoted by ligands that raise the energy, thereby enhancing resonance, of the filled Cu d(xz) orbital that hybridizes with the in-plane pi* orbital of O-2. Conversely, ligands that raise the energy of the filled Cu d(z)(2) orbital foster a preference for end-on binding as this is the only mode that permits good overlap with the in-plane O-2 pi*. Because the overlap of Cu d(z)(2) with O-2 pi* is reduced as compared to the overlap of Cu d(xz) with the same O-2 orbital, the resonance is also reduced, leading to generally more stable triplet states relative to singlets in the end-on geometry as compared to the side-on geometry, where singlet ground states become more easily accessible once ligands are stronger donors. Biradical Cu(II)-O-2 superoxide character in the electronic structure of the supported complexes leads to significant challenges for accurate quantum chemical calculations that are best addressed by exploiting the spin-purified M06L local density functional, single-reference completely renormalized coupled-cluster theory, or multireference second-order perturbation theory, All of which provide predictions that are qualitatively and quantitatively consistent with one another.