SARS-CoV-2 evolves rapidly in part because of its high mutation rate. Here we examine whether this mutational process itself has changed during viral evolution. To do this, we quantify the relative rates of different types of single nucleotide mutations at four-fold degenerate sites in the viral genome across millions of human SARS-CoV-2 sequences. We find clear shifts in the relative rates of several types of mutations during SARS-CoV-2 evolution. The most striking trend is a roughly two-fold decrease in the relative rate of G→T mutations in Omicron versus early clades, as was recently noted by Ruis et al (2022). There is also a decrease in the relative rate of C→T mutations in Delta, and other subtle changes in the mutation spectrum along the phylogeny. We speculate that these changes in the mutation spectrum could arise from viral mutations that affect genome replication, packaging, and antagonization of host innate-immune factors-although environmental factors could also play a role. Interestingly, the mutation spectrum of Omicron is more similar than that of earlier SARS-CoV-2 clades to the spectrum that shaped the long-term evolution of sarbecoviruses. Overall, our work shows that the mutation process is itself a dynamic variable during SARS-CoV-2 evolution, and suggests that human SARS-CoV-2 may be trending towards a mutation spectrum more similar to that of other animal sarbecoviruses.