Three alternative algorithms for dynamic aeroelastic stability analysis are investigated. They are based either on direct search of oscillatory conditions of the coupled system or on eigenvalue analysis from frequency-domain sampling of the unsteady aerodynamics. The aerodynamic forcing is obtained with a harmonic-balanced solver developed for a general-purpose finite-volume fluid dynamics solver. This new implementation is first verified against experimental flutter results from the literature. A wing with relatively low bending stiffness is then used to explore the relative performance of each approach, both in terms of the numerical robustness and of their usability to support aeroelastic design.