We study the temporal stability of the Orr–Sommerfeld and Squire equations in channels with turbulent mean velocity profiles and turbulent eddy viscosities. Friction Reynolds numbers up to $Re_\tau \,{=}\, 2\,{\times}\, 10^4$ are considered. All the eigensolutions of the problem are damped, but initial perturbations with wavelengths $\lambda_x \,{>}\, \lambda_z$ can grow temporarily before decaying. The most amplified solutions reproduce the organization of turbulent structures in actual channels, including their self-similar spreading in the logarithmic region. The typical widths of the near-wall streaks and of the large-scale structures of the outer layer, $\lambda_z^+ \,{=}\, 100$ and $\lambda_z/h \,{=}\, 3$, are predicted well. The dynamics of the most amplified solutions is roughly the same regardless of the wavelength of the perturbations and of the Reynolds number. They start with a wall-normal $v$ event which does not grow but which forces streamwise velocity fluctuations by stirring the mean shear ($uv\,{<}\,0$). The resulting $u$ fluctuations grow significantly and last longer than the $v$ ones, and contain nearly all the kinetic energy at the instant of maximum amplification.