  A number of important observables exhibit logarithms in their perturbative
description that are induced by emissions at widely separated rapidities. These
include transverse-momentum ($q_T$) logarithms, logarithms involving
heavy-quark or electroweak gauge boson masses, and small-$x$ logarithms. In
this paper, we initiate the study of rapidity logarithms, and the associated
rapidity divergences, at subleading order in the power expansion. This is
accomplished using the soft collinear effective theory (SCET). We discuss the
structure of subleading-power rapidity divergences and how to consistently
regulate them. We introduce a new pure rapidity regulator and a corresponding
$\overline{\rm MS}$-like scheme, which handles rapidity divergences while
maintaining the homogeneity of the power expansion. We find that power-law
rapidity divergences appear at subleading power, which give rise to derivatives
of parton distribution functions. As a concrete example, we consider the $q_T$
spectrum for color-singlet production, for which we compute the complete
$q_T^2/Q^2$ suppressed power corrections at $\mathcal{O}(\alpha_s)$, including
both logarithmic and nonlogarithmic terms. Our results also represent an
important first step towards carrying out a resummation of subleading-power
rapidity logarithms.
