Abstract Background and objectives To understand how organisms evolve, it is fundamental to study how mutations emerge and establish. Here, we estimated the rate of mutation accumulation of SARS-CoV-2 in vitro and investigated the repeatability of its evolution when facing a new cell type but no immune or drug pressures. Methodology We performed experimental evolution with two strains of SARS-CoV-2, one carrying the originally described spike protein (CoV-2-D) and another carrying the D614G mutation that has spread worldwide (CoV-2-G). After 15 passages in Vero cells and whole genome sequencing, we characterized the spectrum and rate of the emerging mutations and looked for evidences of selection across the genomes of both strains. Results From the frequencies of the mutations accumulated, and excluding the genes with signals of selection, we estimate a spontaneous mutation rate of 1.3 × 10−6 ± 0.2 × 10−6 per-base per-infection cycle (mean across both lineages of SARS-CoV-2 ± 2SEM). We further show that mutation accumulation is larger in the CoV-2-D lineage and heterogeneous along the genome, consistent with the action of positive selection on the spike protein, which accumulated five times more mutations than the corresponding genomic average. We also observe the emergence of mutators in the CoV-2-G background, likely linked to mutations in the RNA-dependent RNA polymerase and/or in the error-correcting exonuclease protein. Conclusions and implications These results provide valuable information on how spontaneous mutations emerge in SARS-CoV-2 and on how selection can shape its genome toward adaptation to new environments. Lay Summary: Each time a virus replicates inside a cell, errors (mutations) occur. Here, via laboratory propagation in cells originally isolated from the kidney epithelium of African green monkeys, we estimated the rate at which the SARS-CoV-2 virus mutates—an important parameter for understanding how it can evolve within and across humans. We also confirm the potential of its Spike protein to adapt to a new environment and report the emergence of mutators—viral populations where mutations occur at a significantly faster rate.