New and more transmissible variants of SARS-CoV-2 have arisen multiple times over the course of the pandemic. Rapidly identifying mutations that affect transmission could facilitate outbreak control efforts and highlight new variants that warrant further study. Here we develop an analytical path integral model that infers the transmission effects of mutations from genomic surveillance data. Applying our model to SARS-CoV-2 data across many regions, we find multiple mutations that substantially affect the transmission rate, both within and outside the Spike protein. The strongly selected mutations we infer for SARS-CoV-2 are supported by experimental evidence regarding their effects on neutralizing antibody escape, T cell/immune evasion, replication/infectivity, cell entry, and structure/cleavage. Importantly, our model detects lineages with increased transmission even at low frequencies. As an example, we infer significant transmission advantages for the Alpha, Delta, and Omicron variants shortly after their appearances in regional data, when their local frequencies were only around 1-2%. Our model thus facilitates the rapid identification of variants and mutations that affect transmission from genomic surveillance data.