Routing trains through busy railway station layouts is an important part of the timetabling process. For each train, a feasible route has to be determined to provide reliable operations, given the arrival and departure times at stations. In this paper, we propose a model for stable and robust train routing with the goal to minimize capacity occupation and maximize robustness. We define a multi-objective optimization problem and provide the heuristic RouteCare based on a max-plus automata model and a delay propagation model. We consider microscopic infrastructure to guarantee practical feasibility. The performance of the proposed algorithm is demonstrated on real-life instances of the Dutch railway network. The generated solutions outperformed the variants of RouteCare that independently maximize stability or robustness by 10.4% and 9.5%, respectively. In addition, RouteCare showed that even for the same number of resources used, a more robust route plan can be found that uses the station capacity more efficiently.