Clonal populations of cells continuously evolve new genetic diversity, but it takes a significant amount of time for the progeny of a single cell with a new beneficial mutation to outstrip both its ancestor and competitors to fully dominate a population. If these driver mutations can be discovered earlier - while they are still extremely rare - and profiled in large numbers, it may be possible to anticipate the future evolution of similar cell populations. For example, one could diagnose the likely course of incipient diseases, such as cancer and bacterial infections, and better judge which treatments will be effective, by tracking rare drug-resistant variants. To test this approach, we replayed the first 500 generations of a >70,000-generation Escherichia coli experiment and examined the trajectories of new mutations in eight genes known to be under positive selection in this environment in six populations. By employing a deep sequencing procedure using unique molecular identifiers and target enrichment we were able to track 301 beneficial mutations at frequencies as low as 0.01% and infer the fitness effects of 240 of these. Distinct molecular signatures of selection on protein structure and function were evident for the three genes in which beneficial mutations were most common (nadR, pykF, and topA). We detected mutations hundreds of generations before they became dominant and tracked beneficial alleles in genes that were not mutated in the long-term experiment until thousands of generations had passed. This type of targeted adaptome sequencing approach could function as an early warning system to inform interventions that aim to prevent undesirable evolution.