  We investigate the linear stability of shears near the Couette flow for a
class of 2D incompressible stably stratified fluids. Our main result consists
of nearly optimal decay rates for perturbations of stationary states whose
velocities are monotone shear flows $(U(y),0)$ and have an exponential density
profile. In the case of the Couette flow $U(y)=y$, we recover the rates
predicted by Hartman in 1975, by adopting an explicit point-wise approach in
frequency space. As a by-product, this implies optimal decay rates as well as
Lyapunov instability in $L^2$ for the vorticity. For the previously unexplored
case of more general shear flows close to Couette, the inviscid damping results
follow by a weighted energy estimate. Each outcome concerning the stably
stratified regime applies to the Boussinesq equations as well. Remarkably, our
results hold under the celebrated Miles-Howard criterion for stratified fluids.
