  We unravel the linear stability properties of an otherwise stagnant ultrathin
non-wetting liquid film of thickness $h_o$ coating a spherical substrate of
radius $R$. The configuration is known to be unstable due to the competition of
the destabilizing van der Waals (vdW) forces and the stabilizing surface
tension force. The governing equations of motion written in the Stokes limit of
negligible liquid inertia and an accompanying lubrication model are linearised
about the zero-velocity base state and decomposed into normal modes in order to
obtain the temporal dispersion relation. Discrete unstable modes are identified
and tracked as a function of a capillary number $Ca$ measuring the relative
importance of surface tension to vdW forces and the film aspect ratio $\eta =
R/h_o$. For small enough values of the capillary number only the first polar
mode is unstable, and the corresponding maximum growth rate is shown to be a
universal function of $\eta$. Lubrication theory is seen to provide a good
quantitative prediction of the film stability properties for $\eta\gg 1$.
