We consider the occurrence of small axisymmetric pinholes in an otherwise uniform infinite thin liquid film. Corresponding to any particular undisturbed film thickness there exists precisely one (unstable) equilibrium solution reflecting a balance between surface tension and gravity effects. If a pinhole is smaller than this critical size the pinhole tends to close over and "heal". If a pinhole is larger it tends to open out. So determination of this critical hole size is crucial. We examine this problem in the case of a "small" pinhole where the fundamental length-scale in the film is much smaller than the capillary length. Solutions are obtained using matched asymptotic expansions for which several different scalings are necessary.