The dynamics of adaptation determine which mutations fix in a population, and hence how reproducible evolution will be. This is central to understanding the spectra of mutations recovered in the evolution of antibiotic resistance, the response of pathogens to immune selection, and the dynamics of cancer progression. In laboratory evolution experiments, demonstrably beneficial mutations are found repeatedly, but are often accompanied by other mutations with no obvious benefit. Here we use whole-genome whole-population sequencing to examine the dynamics of genome sequence evolution at high temporal resolution in 40 replicate Saccharomyces cerevisiae populations growing in rich medium for 1,000 generations. We find pervasive genetic hitchhiking: multiple mutations arise and move synchronously through the population as mutational 'cohorts'. Multiple clonal cohorts are often present simultaneously, competing with each other in the same population. Our results show that patterns of sequence evolution are driven by a balance between these chance effects of hitchhiking and interference, which increase stochastic variation in evolutionary outcomes, and the deterministic action of selection on individual mutations, which favours parallel evolutionary solutions in replicate populations.