The swimming stability of spermatozoa with a specified planar beat pattern in the presence of a no-slip flat surface is explored in a modelling study exploiting direct numerical computation via the boundary element method and dynamical systems theory. Parameter sweeps varying the sperm head morphology and flagellar beat pattern wavenumber are conducted and reveal that stable surface swimming is a robust hydrodynamical phenomenon across extensive parameter values, emphasising that diverse sperm will readily swim adjacent to a surface without detailed feedback. There is little sensitivity to the details of the sperm head morphologies considered and, in particular, cells with human sperm head geometries are well approximated by those with prolate ellipsoid heads. However, surface accumulation is predicted to be inhibited by changes associated with mammalian sperm hyperactivation and quantitative aspects, such as the accumulation height associated with surface swimming, are sensitive to the flagellar beat pattern wavenumber and even to the asymptotically small modelling approximations of slender body theory. In particular, the predicted sensitivity of the accumulation height of swimming sperm to the beat pattern wavenumber is sufficient to suggest the possibility that the limited focal depth of typical microscopy studies analysing flagellar patterns with a fixed focal plane may inadvertently bias the wavenumber of the sperm that are observed. (C) 2014 Elsevier Ltd. All rights reserved.