We exemplify the rigorous treatment of capillarity-driven free-surface flows by considering the “teapot effect” as an appealing daily-life phenomenon. To this end, we formulate the problem in full, elucidate its structure for small viscous influence in a first step and then refine the results by scrutinising the underlying potential flow in due detail. Finally, we address the subtleties arising when it comes to the rational inclusion of viscous effects so as to sort the real flow out of a one-parametric class of inviscid-flow solutions (so-called selection problem). This approach shall demonstrate the successful, systematic treatment of complex flow problems, involving a variety of disparate length scales. Amongst others, it is demonstrated how the correctly performed abstraction process can unveil unexpected mechanisms and deepen the understanding of known physical phenomena. Many intriguing questions associated with the effect in focus, but also neighbouring fields, are found as not settled conclusively yet. This calls for further analytical and numerical progress.