Epithelial folding is crucial to the formation of tissues during development and occurs via a variety of mechanisms. In some tissues, folding is mediated by the presence of actomyosin-rich cables: supracellular structures that are commonly found at compartmental boundaries. However, actomyosin-rich cables in other contexts do not facilitate tissue folding, and so the relationship between the two remains unclear. Now, Bénédicte Sanson and colleagues provide evidence for a new concept, which is that epithelial tissues have specific mechanisms in place to suppress fold formation at actomyosin-rich boundaries or structures. The authors show that actomyosin contractility is required for fold formation at the parasegmental boundary in the developing Drosophila embryo, but that, unexpectedly, specific pathways control the degree of epithelial folding, downstream of opposing regulation from the Wingless and Hedgehog signalling pathways. Key factors involved are the Myosin II phosphatase Flapwing and the homologue of Par-3, Bazooka (Baz). Depletion of Baz at boundary cell-cell contacts correlates with fold suppression, while an increase in Baz promotes excessive folding specifically at actomyosin-rich boundaries. The activity of Baz/Par-3 in groove formation differs from its previously reported role in junction lowering during dorsal fold formation, pointing towards a novel role of Baz/Par-3.