Molecular evolution can be conceptualized as a walk over a “fitness landscape”, or the function of fitness (e.g., catalytic activity) over the space of all possible sequences. Understanding evolution requires knowing the structure of the fitness landscape and identifying the viable evolutionary pathways through the landscape. However, the fitness landscape for any catalytic biomolecule is largely unknown. The evolution of catalytic RNA is of special interest because RNA is believed to have been foundational to early life. In particular, an essential activity leading to the genetic code would be the reaction of ribozymes with activated amino acids, such as 5(4H)-oxazolones, to form aminoacyl-RNA. Here we combine in vitro selection with a massively parallel kinetic assay to map a fitness landscape for self-aminoacylating RNA, with nearly complete coverage of sequence space in a central 21-nucleotide region. The method (SCAPE: sequencing to measure catalytic activity paired with in vitro evolution) shows that the landscape contains three major ribozyme families (landscape peaks). An analysis of evolutionary pathways shows that, while local optimization within a ribozyme family would be possible, optimization of activity over the entire landscape would be frustrated by large valleys of low activity. The sequence motifs associated with each peak represent different solutions to the problem of catalysis, so the inability to traverse the landscape globally corresponds to an inability to restructure the ribozyme without losing activity. The frustrated nature of the evolutionary network suggests that chance emergence of a ribozyme motif would be more important than optimization by natural selection.