Allele age has long been a focus of population genetic research, primarily because it can be an important clue to the fitness effects of an allele. By virtue of their effects on fitness, alleles under directional selection are expected to be younger than neutral alleles of the same frequency. We developed a new coalescent-based estimator of a close proxy for allele age, the time when a copy of an allele first shares common ancestry with other chromosomes in a sample not carrying that allele. The estimator performs well, including for the very rarest of alleles that occur just once in a sample, with a bias that is typically negative. The estimator is mostly insensitive to population demography and to factors that can arise in population genomic pipelines, including the statistical phasing of chromosomes. Applications to 1000 Genomes Data and UK10K genome data confirm predictions that singleton alleles that alter proteins are significantly younger than those that do not, with a greater difference in the larger UK10K dataset, as expected. The 1000 Genomes populations varied markedly in their distributions for singleton allele ages, suggesting that these distributions can be used to inform models of demographic history, including recent events that are only revealed by their impacts on the ages of very rare alleles.