A key objective of population genomics is to identify portions of the genome that have been shaped by natural selection rather than by neutral divergence. A previously recognized but underappreciated challenge to this objective is that observations of allele frequencies across genomes in natural populations often correspond to a single, unreplicated instance of the outcome of evolution. This is because the composition of each individual genomic region and population is expected to be the outcome of a unique array of evolutionary processes. Given a single observation, inference of the evolutionary processes that led to the observed state of a locus is associated with considerable uncertainty. This constraint on inference can be ameliorated by utilizing multi-allelic (e.g. DNA haplotypes) rather than bi-allelic markers, by analysing two or more populations with certain models and by utilizing studies of replicated experimental evolution. Future progress in population genomics will follow from research that recognizes the ‘n = 1 constraint’ and that utilizes appropriate and explicit evolutionary models for analysis.