This paper presents the results of the dynamic stability analysis of a flexible spinning launch vehicle subjected to thrust modeled as a thin-walled composite beam with circular cross section. Due to the presence of gyroscopic forces, we mainly aimed to find divergence and/or flutter instabilities and establish the stability boundaries of the spinning beam. For this purpose, we implemented a circumferentially uniform stiffness (CUS) layup configuration to exhibit the coupled motion of bending-bending-shear. The solution of the eigenvalue problem is handled by the extended Galerkin method (EGM), and we computed the results addressing the effects of various parameters such as spin speed, axial load, ply angle, aspect ratio and transverse shear on the dynamic stability of the beam. Insights revealed by this study contribute to the design of advanced aerospace structures modeled as thin-walled composite beams reflecting the potential influence of transverse shear and aspect ratio on dynamic stability characteristics. A notable contribution is that we show that divergence/flutter instabilities can be delayed or even avoided using the directionality property of composite materials.