We develop a new method to locally cluster curves and discover functional motifs, that is, typical shapes that may recur several times along and across the curves capturing important local characteristics. In order to identify these shared curve portions, our method leverages ideas from functional data analysis (joint clustering and alignment of curves), bioinformatics (local alignment through the extension of high similarity seeds) and fuzzy clustering (curves belonging to more than one cluster, if they contain more than one typical shape). It can employ various dissimilarity measures and incorporate derivatives in the discovery process, thus exploiting complex facets of shapes. We demonstrate the performance of our method with an extensive simulation study, and show how it generalizes other clustering methods for functional data. Finally, we provide real data applications to Italian Covid-19 death curves and Omics data related to mutagenesis.