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The prevalence of neutral mutations implies that biological systems typically have many more genotypes than phenotypes. But, can the way that genotypes are distributed over phenotypes determine evolutionary outcomes? Answering such questions is difficult, in part because the number of genotypes can be hyper-astronomically large. By solving the genotype-phenotype (GP) map for RNA secondary structure (SS) for systems up to length L = 126 nucleotides (where the set of all possible RNA strands would weigh more than the mass of the visible universe), we show that the GP map strongly constrains the evolution of non-coding RNA (ncRNA). Simple random sampling over genotypes predicts the distribution of properties such as the mutational robustness or the number of stems per SS found in naturally occurring ncRNA with surprising accuracy. Because we ignore natural selection, this strikingly close correspondence with the mapping suggests that structures allowing for functionality are easily discovered, despite the enormous size of the genetic spaces. The mapping is extremely biased: the majority of genotypes map to an exponentially small portion of the morphospace of all biophysically possible structures. Such strong constraints provide a non-adaptive explanation for the convergent evolution of structures such as the hammerhead ribozyme. These results present a particularly clear example of bias in the arrival of variation strongly shaping evolutionary outcomes and may be relevant to Mayr's distinction between proximate and ultimate causes in evolutionary biology.

The structure of the genotype–phenotype map strongly constrains the evolution of non-coding RNA