Stability of localized solutions arising in a fourth–order differential equation modelling struts is investigated. It was shown by Buffoni et al. in 1996 that the model exhibits many multimodal buckling states bifurcating from a primary buckling mode. In this article, using analytical and numerical techniques, the primary mode is shown to be unstable under dead loading for all axial loads, while it is likely to be stable under rigid loading for small axial loads. Furthermore, for general reversible or conservative systems, stability of the multimodal solutions is established assuming stability of the primary state. Since this hypothesis is not satisfied for the buckling mode arising in the strut model, any multimodal buckling state will be unstable under dead and rigid loading.