We consider the behaviour of buckling-driven thin-film blisterings using Von Karman's plate theory. Our results are asymptotic in the thinness and emphasize the incorporation of in-plane displacements in the model, a factor often ignored in the literature. Our work indicates that the inclusion of these displacements has a profound effect on the nature of solutions. We compare different constraints on displacements by estimating the leading order of corresponding energies. Our results strongly suggest that branching and other fine-scale structures arise in thin-film blistering, as has been observed in experiments.