In the absence of effective treatment, COVID-19 is likely to remain a global disease burden. Compounding this threat is the near certainty that novel coronaviruses with pandemic potential will emerge in years to come. Pan-coronavirus drugs - agents active against both SARS-CoV-2 and other coronaviruses - would address both threats. A strategy to develop such broad-spectrum inhibitors is to pharmacologically target binding sites on SARS-CoV-2 proteins that are highly conserved in other known coronaviruses, the assumption being that any selective pressure to keep a site conserved across past viruses will apply to future ones. Here, we systematically mapped druggable binding pockets on the experimental structure of fifteen SARS-CoV-2 proteins and analyzed their variation across twenty-seven - and {beta}-coronaviruses and across thousands of SARS-CoV-2 samples from COVID-19 patients. We find that the two most conserved druggable sites are a pocket overlapping the RNA binding site of the helicase nsp13, and the catalytic site of the RNA-dependent RNA polymerase nsp12, both components of the viral replication-transcription complex. We present the data on a public web portal (https://www.thesgc.org/SARSCoV2_pocketome/) where users can interactively navigate individual protein structures and view the genetic variability of drug binding pockets in 3D.