  This paper concerns spectral instability of shear flows in the incompressible
Navier-Stokes equations with sufficiently large Reynolds number: $R\to \infty$.
It is well-documented in the physical literature, going back to Heisenberg,
C.C. Lin, Tollmien, Drazin and Reid, that generic plane shear profiles other
than the linear Couette flow are linearly unstable for sufficiently large
Reynolds number. In this work, we provide a complete mathematical proof of
these physical results. In the case of a symmetric channel flow, our analysis
gives exact unstable eigenvalues and eigenfunctions, showing that the solution
could grow slowly at the rate of $e^{t/\sqrt {\alpha R}}$, where $\alpha$ is
the small spatial frequency that remains between lower and upper marginal
stability curves: $\alpha_\mathrm{low}(R) \approx R^{-1/7}$ and
$\alpha_\mathrm{up}(R) \approx R^{-1/11}$. We introduce a new, operator-based
approach, which avoids to deal with matching inner and outer asymptotic
expansions, but instead involves a careful study of singularity in the critical
layers by deriving pointwise bounds on the Green function of the corresponding
Rayleigh and Airy operators.
