Geometric instabilities in bilayered structures control the surface morphology in a wide range of biological and technical systems. Depending on the application, different mechanisms induce compressive stresses in the bilayer. However, the impact of the chosen origin of compression on the critical conditions, post-buckling evolution and higher-order pattern selection remains insufficiently understood. Here, we conduct a numerical study on a finite-element set-up and systematically vary well-known factors contributing to pattern selection under the four main origins of compression: film growth, substrate shrinkage and whole-domain compression with and without pre-stretch. We find that the origin of compression determines the substrate stretch state at the primary instability point and thus significantly affects the critical buckling conditions. Similarly, it leads to different post-buckling evolutions and secondary instability patterns when the load further increases. Our results emphasize that future phase diagrams of geometric instabilities should incorporate not only the film thickness but also the origin of compression. Thoroughly understanding the influence of the origin of compression on geometric instabilities is crucial to solving real-life problems such as the engineering of smart surfaces or the diagnosis of neuronal disorders, which typically involve temporally or spatially combined origins of compression.