We formulate some first fundamental elements of an approach for assessing the logarithmic accuracy of parton-shower algorithms based on two broad criteria: their ability to reproduce the singularity structure of multi-parton matrix elements, and their ability to reproduce logarithmic resummation results. We illustrate our approach by considering properties of two transverse-momentum ordered final-state showers, examining features up to second order in the strong coupling. In particular we identify regions where they fail to reproduce the known singular limits of matrix elements. The characteristics of the shower that are responsible for this also affect the logarithmic resummation accuracies of the shower, both in terms of leading (double) logarithms at subleading colour number and next-to-leading (single) logarithms at leading colour number.