We explore minimizers of a simple functional modeling the free energy of thin liquid layers over a solid substrate under the combined effect of surface, gravitational, and intermolecular potentials. When the latter ones have a mild singularity at short ranges, minimizers are compactly supported and display a right microscopic contact angle. In addition, depending on the form of the potential, the macroscopic shape can either be droplet-like or pancake-like, with a transition profile between the two (at zero spreading coefficient). These results generalize, complete, and give mathematical rigor to de Gennes' formal discussion of spreading equilibria. Uniqueness and non-uniqueness phenomena are also discussed. This is a joint work with Riccardo Durastanti.