The stability properties of the rotating free surface flow in a cylindrical container isstudied using a global stability approach, considering successively three models. For thecase of solid body rotation (Newton’s bucket), all eigenmodes are found to be stable, andare classified into three families: gravity waves, singular inertial modes, and Rossbywaves. For the case of a potential flow, an instability is found. The mechanism isexplained as a resonance between gravity waves and centrifugal waves, and is thought to beat the origin of the “rotating polygon instability” observed in experiments where the flowis driven by rotation of the bottom plate (see L. Tophøj, J. Mougel, T. Bohr, D. Fabre,The Rotating Polygon Instability of a Swirling Free Surface Flow, Phys. Rev. Lett. 110(2013) 194502). Finally, in the case of the Rankine vortex which in fact consists in thecombination of the two first cases, we report a new instability mechanism involving Rossbyand gravity waves.