Subcritical bifurcations—where the post-critical branch emanating from a bifurcation is unstable—are a particularly pernicious failure mode for nonlinear structures. In the case of buckling, perturbations applied to the structure, for example by means of geometric or loading imperfections, lead to instability at lower loads than an analysis of the perfect structure would suggest. As a result, it is important that the sensitivity of the structure to perturbations and pre-mature instability is accurately assessed. One possibility is to propagate uncertainty through a model to quantify the possible worst-case buckling load, with the associated high computational expense that such a stochastic analysis entails. Another approach is to analyse the perfect structure and to find edge states (saddle points with one unstable direction) in the energy basin boundary surrounding the stable pre-buckling state. Edge states with particularly low energy barrier can then be identified as likely escape routes from the pre-buckling state in the presence of static or dynamic perturbations, and serve as targeted modes to investigate in a sensitivity study. Such an approach has recently been applied successfully to the study of axially compressed cylindrical shells, where various unstable dimple solutions appear as edge states in the basin boundary of the stable pre-buckling state. While various edge state algorithms exist in the literature (e.g. dimer or conjugate-peak refinement ), few are tailored to application within the finite element method, where the Jacobian (tangent stiffness matrix) is readily available. Here, I present new one-sided and two-sided saddle point algorithms (requiring one or two known stable states) based on deflation. In deflation, known equilibria are removed from the solution space to guide the iterative Newton solver towards a new solution. If a deflation algorithm is initiated from a known stable equilibrium, there is, however, no control over which solution (stable, saddle, or higher-order saddle) is found. This is particularly important for optimised and ultra-lightweight structures where the basin boundary of the pre-buckling state usually contains multiple unstable edge states. The key novelty of the present approach is that the solver is first guided into the vicinity of a saddle point such that the deflation solver reliably converges to this solution.