Extensive global sampling and sequencing of the pandemic virus 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 characterise within-host diversity and transmission we deep-sequenced 1313 clinical samples from the UK. SARS-CoV-2 infections are characterised by low levels of within-host diversity when viral loads are high, and 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 which are readily observable on the phylogenetic tree. Our results suggest that transmission-enhancing and/or immune-escape variants are likely to arise infrequently, but could spread rapidly if successfully transmitted.