  We analyze the decay and instant regularization properties of the evolution
semigroups generated by two-dimensional drift-diffusion equations in which the
scalar is advected by a shear flow and dissipated by full or partial diffusion.
We consider both the space-periodic $\mathbb{T}^2$ setting and the case of a
bounded channel $\mathbb{T}\times [0,1]$ with no-flux boundary conditions. In
the infinite P\'eclet number limit (diffusivity $\nu\to 0$), our work
quantifies the enhanced dissipation effect due to the shear. We also obtain
hypoelliptic regularization, showing that solutions are instantly Gevrey
regular even with only partial diffusion. The proofs rely on localized spectral
gap inequalities and ideas from hypocoercivity with an augmented energy
functional with weights replaced by pseudo-differential operators (of a rather
simple form). As an application, we study small noise inviscid limits of
invariant measures of stochastic perturbations of passive scalars, and show
that the classical Freidlin scaling between noise and diffusion can be
modified. In particular, although statistically stationary solutions blow up in
$H^1$ in the limit $\nu\to0$, we show that viscous invariant measures still
converge to a unique inviscid measure.
