SARS-CoV-2 continues to spread and infect people worldwide. While most effort into characterizing variants of this virus have focused on non-synonymous changes, accumulation of synonymous mutations in different viral variants has also occurred. Here we characterize six Variants of Concern in terms of their mutational content, and make predictions regarding the impact of those mutations on potential genomic RNA secondary structure and stability. Our hypothesis is that if non-protein changing, yet RNA structure-changing mutations impact viral fitness by imposing deleterious change to predicted RNA structure, we would expect to those mutations to be less abundant, while if those synonymous mutations do not impact viral fitness through influence of RNA structure, we would see them more frequently than non-synonymous mutations. We find that synonymous mutations typically have no or modest impact to RNA secondary structure. As synonymous mutations are free from the selective pressure imposed on protein-altering mutations, the impact of synonymous mutations is largely limited to RNA secondary structure considerations. The absence of major, structure-altering synonymous mutations emphasize the importance of RNA structure, including within coding regions, to viral fitness. Synonymous mutations should be included in the characterization of emerging RNA viruses as these mutations may confer effects to viral fitness via RNA secondary structural modifications.