Single-stranded DNA (ssDNA) viruses have genomes that are potentially capable of forming complex secondary structures through Watson-Crick base pairing between their constituent nucleotides. A few of the structural elements formed by such base pairings are, in fact, known to have important functions during the replication of many ssDNA viruses. Unknown, however, are (i) whether numerous additional ssDNA virus genomic structural elements predicted to exist by computational DNA folding methods actually exist and (ii) whether those structures that do exist have any biological relevance. We therefore computationally inferred lists of the most evolutionarily conserved structures within a diverse selection of animal- and plant-infecting ssDNA viruses drawn from the families Circoviridae, Anelloviridae, Parvoviridae, Nanoviridae, and Geminiviridae and analyzed these for evidence of natural selection favoring the maintenance of these structures. While we find evidence that is consistent with purifying selection being stronger at nucleotide sites that are predicted to be base paired than at sites predicted to be unpaired, we also find strong associations between sites that are predicted to pair with one another and site pairs that are apparently coevolving in a complementary fashion. Collectively, these results indicate that natural selection actively preserves much of the pervasive secondary structure that is evident within eukaryote-infecting ssDNA virus genomes and, therefore, that much of this structure is biologically functional. Lastly, we provide examples of various highly conserved but completely uncharacterized structural elements that likely have important functions within some of the ssDNA virus genomes analyzed here.