Supply networks need to exhibit stability in order to remain functional. Here, we apply a generalized modeling (GM) approach, which has a strong pedigree in the analysis of dynamical systems, to study the stability of real‐world supply networks. It goes beyond purely structural network analysis approaches by incorporating material flows, which are defining characteristics of supply networks. The analysis focuses on the network of interactions between material flows, providing new conceptualizations to capture key aspects of production and inventory policies. We provide stability analyses of two contrasting real‐world networks—that of an industrial engine manufacturer and an industry‐level network in the luxury goods sector. We highlight the criticality of links with suppliers that involve the dispatch, processing, and return of parts or sub‐assemblies, cyclic motifs that involve separate paths from a common supplier to a common firm downstream, and competing demands of different end products at specific nodes. Based on a critical discussion of our findings in the context of the supply chain management literature, we generate five propositions to advance knowledge and understanding of supply network stability. We discuss the implications of the propositions for the effective management, control, and development of supply networks. The GM approach enables fast screening to identify hidden vulnerabilities in extensive supply networks.