Buckling of the initial momentumless stressed state of a thin shell caused by weak fixing of one of its edges is considered. In this case the buckling form is localized in the vicinity of the given edge. Eight versions of boundary conditions, which for some parameter values of the problem admit the occurrence of such buckling forms along with the critical load reduction at the same time, are found. The stability of a cylindrical shell or a panel with a loosely fixed curvilinear (or cylindrical) edge under axial compression and that of a buckling shell of rotation under axial compression are considered.