  When the allowed range of an observable grows order-by-order in perturbation
theory, its perturbative expansion can have discontinuities (as in the $C$
parameter) or discontinuities in its derivatives (as in thrust or heavy jet
mass) called Sudakov shoulders. We explore the origin of these logarithms using
both perturbation theory and effective field theory. We show that for thrust
and heavy jet mass, the logarithms arise from kinematic configurations with
narrow jets and deduce the next-to-leading logarithmic series. The
left-shoulder logarithms in heavy jet mass ($\rho$) of the form $r\ \alpha_s^n
\ln^{2n}r $ with $r=\frac{1}{3}-\rho$ are particularly dangerous, because they
invalidate fixed order perturbation theory in regions traditionally used to
extract $\alpha_s$. Although the factorization formula shows there are no
non-global logarithms, we find Landau-pole like singularities in the resummed
distribution associated with the cusp anomalous dimension, and that power
corrections are exceptionally important.
