  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 $N_C$ and next-to-leading (single) logarithms at leading $N_C$.
