The dorsal, anal and caudal fins of vertebrates are proposed to have originated by the partitioning and transformation of the continuous median fin fold that is plesiomorphic to chordates. Evaluating this hypothesis has been challenging, because it is unclear how the median fin fold relates to the adult median fins of vertebrates. To understand how new median fins originate, here we study the development and diversity of adipose fins. Phylogenetic mapping shows that in all lineages except Characoidei (Characiformes) adipose fins develop from a domain of the larval median fin fold. To inform how the larva's median fin fold contributes to the adipose fin, we study Corydoras aeneus (Siluriformes). As the fin fold reduces around the prospective site of the adipose fin, a fin spine develops in the fold, growing both proximally and distally, and sensory innervation, which appears to originate from the recurrent ramus of the facial nerve and from dorsal rami of the spinal cord, develops in the adipose fin membrane. Collectively, these data show how a plesiomorphic median fin fold can serve as scaffolding for the evolution and development of novel, individuated median fins, consistent with the median fin fold hypothesis.