We examine the dynamics of a thin film of viscous fluid on the inside surface of a cylinder with horizontal axis, rotating about this axis. Using the so-called lubrication approximation, we derive an asymptotic equation for three-dimensional motion of the film and use this equation to examine its linear stability. It is demonstrated that: (i) there are infinitely many normal modes (harmonic in the axial variable and time), which are all neutrally stable and their eigenfunctions form a complete set; (ii) but the film is nonetheless unstable with respect to non-harmonic disturbances, which develop singularities in a finite time.