Repeated emergence of SARS-CoV-2 variants with increased fitness necessitates rapid detection and characterization of new lineages. To address this need, we developed PyR0, a hierarchical Bayesian multinomial logistic regression model that infers relative prevalence of all viral lineages across geographic regions, detects lineages increasing in prevalence, and identifies mutations relevant to fitness. Applying PyR0 to all publicly available SARS-CoV-2 genomes, we identify numerous substitutions that increase fitness, including previously identified spike mutations and many non-spike mutations within the nucleocapsid and nonstructural proteins. PyR0 forecasts growth of new lineages from their mutational profile, identifies viral lineages of concern as they emerge, and prioritizes mutations of biological and public health concern for functional characterization.