  In this note, we investigate the stability property of shear flows under the
2D stationary Navier-Stokes equations, and we obtain that the Couette flow
$(y,0)$ is stable under the space of $\mathcal{D}^{1,q}(\mathbb{R}^2)$ for any
$1<q<\infty$ and unstable in the space of
$\mathcal{D}^{1,\infty}(\mathbb{R}^2)$. A key observation is the anisotropic
cut-off function. We also consider the Poiseuille flow $(y^2,0)$, which is
stable in $\mathcal{D}^{1,q}(\mathbb{R}^2)$ with $\frac43<q\leq4.$
