Conical shell theory and piston theory aerodynamics are used to study the aeroelastic stability of the thermal protection system (TPS) on the NASA Hypersonic Inflatable Aerodynamic Decelerator (HIAD). Structural models of the TPS consist of single or multiple orthotropic conical shell systems resting on several circumferential linear elastic supports.The shells in each model may have pinned (simply-supported) or elastically-supported edges. The Lagrangian is formulated in terms of the generalized coordinates for all displacements and the Rayleigh-Ritz method is used to derive the equations of motion. The natural modes of vibration and aeroelastic stability boundaries are found by calculating the eigenvalues and eigenvectors of a large coefficient matrix. When the in-flight configuration of the TPS is approximated as a single shell without elastic supports, asymmetric flutter in many circumferential waves is observed. When the elastic supports are included, the shell flutters symmetrically in zero circumferential waves. Structural damping is found to be important in this case. Aeroelastic models that consider the individual TPS layers as separate shells tend to flutter asymmetrically at high dynamic pressures relative to the single shell models. Several parameter studies also examine the effects of tension, orthotropicity, and elastic support stiffness. Nomenclature a n , b n , c n Modal coordinates for the three shell displacements, m D ef f Effective material bending stiffness, P a m 3 D y , D θ Shell Bending stiffness in the y and θ directions, respectively, P a m 3 D yθ Shell in-plane twisting stiffness, P a m 3 E Material Young's Modulus, P a E y , E θ Shell Young's Modulus in the y and θ directions, respectively, P a fFrequency, Hz f crit Critical frequency, at the flutter or divergence boundary, Hz G yθ Shear modulus, P a h Shell thickness, m k Circumferential wavenumber k crit Critical circumferential wavenumber, at the flutter or divergence boundary K s Spring stiffness for the circumferential elastic supports, P a K P Y R Pyrogel spring stiffness, P a/m m Shell mass per area, kg/m 2