We consider the stability of a thin transversally isotropic circular cylindrical shell under axial compression. We use a local approach according to which the buckling deflection is sought in the form of a doubly periodic function of curvilinear coordinates and the boundary conditions are ignored. We compare the solutions obtained according to the two-dimensional Kirchhoff-Love (KL) and Timoshenko-Reissner (TR) models with the solutions constructed according to the three-dimensional theory. Attention is mainly paid to the case of very small shear stiffness in the transverse direction.