Jet substructure observables play a central role at the Large Hadron Collider for identifying the boosted hadronic decay products of electroweak scale resonances. The complete description of these observables requires understanding both the limit in which hard substructure is resolved, as well as the limit of a jet with a single hard core. In this paper we study in detail the perturbative structure of two prominent jet substructure observables, $N$-subjettiness and the energy correlation functions, as measured on background QCD jets. In particular, we focus on the distinction between the limits in which two-prong structure is resolved or unresolved. Depending on the choice of subjet axes, we demonstrate that at fixed order, $N$-subjettiness can manifest myriad behaviors in the unresolved region: smooth tails, end-point singularities, or singularities in the physical region. The energy correlation functions, by contrast, only have non-singular perturbative tails extending to the end point. We discuss the effect of hadronization on the various observables with Monte Carlo simulation and demonstrate that the modeling of these effects with non-perturbative shape functions is highly dependent on the $N$-subjettiness axes definitions. Our study illustrates those regions of phase space that must be controlled for high-precision jet substructure calculations, and emphasizes how such calculations can be facilitated by designing substructure observables with simple singular structures.

17 pages, 14 figures; v2 PRD version, added table in conclusions listing endpoint behavior of N-subjettiness