I will discuss some simplified models for the shape of liquid droplets on rough solid surfaces, especially Bernoulli-type free boundary problems. In these models small scale roughness leads to large scale non-uniqueness, hysteresis, and anisotropies. In technical terms we need to understand laminating/foliating families of plane-like solutions, this is related to ideas of Aubry-Mather theory, but, unlike most results in that area, we need to consider local (but not global) energy minimizers.