With recent successes in machine learning and artificial intelligence, ever bigger tasks get into focus of artificial-intelligence research. It seems that soon, many tasks in engineering might be dealt with using purely data-driven approaches, gradually replacing domain experts' systematic insights. Increasingly, the only limit seems to be the availability of enough training data and computation power. However, problems in engineering and systems science can have properties that challenge today's popular machine-learning approaches in other ways than by being of large dimensionality regarding parameters, decision variables, and datasets. This contribution presents arguments as to why the control of nonholonomic systems may present such a problem class by posing surprising challenges even at low dimensionality, e.g., when parking nonholonomic vehicles and robots with millimeter-level accuracy. To that end, first, the inherent intricacies of nonholonomic systems are recapitulated as they arise from the underlying sub-Riemannian geometry. Among other things, this includes the role of first-order approximations and the difficulty of faithfully measuring distance for nonholonomic systems. It is then studied what these intricacies mean for model-based feedback control methods, in particular for model predictive control. Based on these geometric and system-theoretic preliminaries, the main part of the contribution explores what this means for data-driven control. Mainly, two flavors of data-driven control are explored. In the first, the idea is to, devoid of any systematic a-priori or expert knowledge, learn a model from measured data, and use said model in a model-based control approach. In the second, the intermediate system-identification step is circumvented by using reinforcement learning. The contribution shows how nonholonomic systems' characteristic intricacies translate to difficulties experienced with these two ways of proceeding. In particular, it is discussed to which extent workable data-driven solutions still require the expert insight they, for many, promise to make obsolete.