We calculate the nonlinear magneto-optical response of free-standing fcc (001), (110), and (111) oriented Fe monolayers. The band structures are determined from first principles using a full-potential linear augmented plane wave method with the additional implementation of spin-orbit coupling. The variation of the spin-orbit coupling strength and the nonlinear magneto-optical spectra upon layer orientation are investigated. We find characteristic differences that indicate an enhanced sensitivity of nonlinear magneto-optics to surface orientation and variation of the in-plane lattice constants. In particular, the crossover from one-dimensional stripe structures to two-dimensional films of (111) layers exhibits a clean signature in the nonlinear Kerr spectra and demonstrates the versatility of nonlinear magneto-optics as a tool for in situ thin-film analysis.