Wrinkling phenomena of stiff thin films on compliant substrates are investigated based on a non-linear finite element model. The resulting non-linear equations are then solved by the Asymptotic Numerical Method (ANM) that gives interactive access to semi-analytical equilibrium branches, which offers considerable advantage of reliability compared with classical iterative algorithms. Bifurcation points are detected through computing bifurcation indicators well adapted to the ANM. The effect of boundary conditions and material properties of the substrate on the bifurcation portrait is carefully studied. The evolution of wrinkling patterns and post-bifurcation modes including period-doubling has been observed beyond the onset of the primary sinusoidal wrinkling mode in the post-buckling range. Display omitted •A classical film/substrate model is analysed by advanced numerical methods.•Dimensional analysis of the model is performed.•Generic bifurcation schemes are established.•The effect of boundary conditions and material properties is investigated.•Bifurcation points on non-linear response curves and post-buckling modes including period-doubling are captured.