Population structure influences genealogical patterns, however data pertaining to how populations are structured are often unavailable or not directly observable. Inference of population structure is highly important in molecular epidemiology where pathogen phylogenetics is increasingly used to infer transmission patterns and detect outbreaks. Discrepancies between observed and idealised genealogies, such as those generated by the coalescent process, can be quantified, and where significant differences occur, may reveal the action of natural selection, host population structure, or other demographic and epidemiological heterogeneities. We have developed a fast non-parametric statistical test for detection of cryptic population structure in time-scaled phylogenetic trees. The test is based on contrasting estimated phylogenies with the theoretically expected phylodynamic ordering of common ancestors in two clades within a coalescent framework. These statistical tests have also motivated the development of algorithms which can be used to quickly screen a phylogenetic tree for clades which are likely to share a distinct demographic or epidemiological history. Epidemiological applications include identification of outbreaks in vulnerable host populations or rapid expansion of genotypes with a fitness advantage. To demonstrate the utility of these methods for outbreak detection, we applied the new methods to large phylogenies reconstructed from thousands of HIV-1 partial pol sequences. This revealed the presence of clades which had grown rapidly in the recent past, and was significantly concentrated in young men, suggesting recent and rapid transmission in that group. Furthermore, to demonstrate the utility of these methods for the study of antimicrobial resistance, we applied the new methods to a large phylogeny reconstructed from whole genome Neisseria gonorrhoeae sequences. We find that population structure detected using these methods closely overlaps with the appearance and expansion of mutations conferring antimicrobial resistance.