This paper introduces topology finding of patterns for shell structures such as beam grids for gridshells or voussoir tessellations for vaults, among others. The authors refer to topology finding, by analogy and in complement to form finding, as the design of the connectivity of these patterns in order to follow architectural, structural and construction requirements. This paper presents a computational approach relying on a specific design space and data structures based on singularity meshes, which encode the information about the singularities in patterns. The designed patterns are structured, i.e. with a low number of singularities, can include high-valency pole points, and respect alignment to surfaces, curves and points. A feature-based exploration approach is introduced with a generation procedure for singularity meshes following the boundaries of a surface as well as point and curve features, using a topological skeleton or medial axis. These features can stem from statics heuristics, whose efficiency is assessed in a case study. A rule-based editing approach for singularity meshes supplements feature-based topology finding, using a grammar of strip rules as parameters to further explore the singularity design space. This conceptual design approach and its algorithms are an aid for topological exploration of patterns for shell-like structures by architects and engineers.