The present paper studies a set of bifurcation paths for a cylindrical shell under external pressure and under axial compression. For both cases, catastrophe theory methods apply; for cylindrical shell subjected to external pressure, buckling fits the wigwam catastrophe; for axially-compressed cylindrical shell, a set of post-buckling solutions does not fit a single catastrophe manifold, but rather comprises a number of high-order umbilical catastrophes. For the first case, five essential parameters, that is parameters that determine buckling load and mode, were revealed.