  We find solutions that describe the levelling of a thin fluid film,
comprising a non-Newtonian power-law fluid, that coats a substrate and evolves
under the influence of surface tension. We consider the evolution from both
periodic and localised initial conditions as separate cases. Particular
(similarity) solutions in each of these two cases exhibit the generic property
that the profiles are weakly singular (that is, higher-order derivatives do not
exist), at points where the pressure gradient vanishes. Numerical simulations
of the thin film equation, with either periodic or localised initial condition,
are shown to approach the appropriate particular solution.
