The possible emergence of a localized buckled configuration for a simple supported linear elastic beam on an elastic foundation subject to axial compression is analyzed numerically. Beams on a softening foundation and, for comparison purposes, beams on a linear elastic foundation are considered. The critical bifurcation mode involves periodic lateral displacements for both foundations. Calculations are carried out for imperfections that can have a component in the shape of the critical periodic bifurcation mode as well as a component with a localized shape. For beams on a softening foundation, the buckled configuration obtained in the calculations depends on the magnitudes of the periodic and localized imperfection components: (i) for a wide range of imperfections, a rather abrupt growth of localized lateral displacements occurs after a load maximum, even if the imperfection magnitudes are such that the critical periodic mode is suppressed; or (ii) if the magnitude of the periodic imperfection component is sufficiently large, the buckled configuration with a localized imperfection involves lateral displacements that are oscillatory but not periodic. For beams on a linear elastic foundation, a calculation with a sufficiently large magnitude localized imperfection gives a localized buckled configuration under increasing load. For beams on both types of foundation, the main features of the lateral displacements in the localized buckled configuration are similar, for example, involving both positive and negative lateral displacements even if the localized imperfection only involves non-negative deflections. •A beam on a softening foundation with periodic and localized imperfections.•A localized buckled shape can occur by a bifurcation after a load maximum.•A localized buckled shape can occur without any periodic imperfection.•A large periodic imperfection acts to suppress emergence of the localized shape.•A localized buckled shape can develop with no softening under increasing load.