  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.
