Extensive global sampling and sequencing of the pandemic virus severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) have enabled researchers to monitor its spread and to identify concerning new variants. Two important determinants of variant spread are how frequently they arise within individuals and how likely they are to be transmitted. To characterize within-host diversity and transmission, we deep-sequenced 1313 clinical samples from the United Kingdom. SARS-CoV-2 infections are characterized by low levels of within-host diversity when viral loads are high and by a narrow bottleneck at transmission. Most variants are either lost or occasionally fixed at the point of transmission, with minimal persistence of shared diversity, patterns that are readily observable on the phylogenetic tree. Our results suggest that transmission-enhancing and/or immune-escape SARS-CoV-2 variants are likely to arise infrequently but could spread rapidly if successfully transmitted.