Let X be a smooth quartic surface not containing lines, defined over a number field κ. We prove that there are only finitely many bitangents to X which are defined over κ. This result can be interpreted as saying that a certain surface, having vanishing irregularity, contains only finitely many rational points. In our proof, we use the geometry of lines of the quartic double solid associated to X. In a somewhat opposite direction, we show that on any quartic surface X over a number field κ, the set of algebraic points in X(κ) which are quadratic over a suitable finite extension κ' of κ is Zariski-dense.
