We study chemical patterns arising from instabilities in reaction-diffusion-advection systems under the influence of shear flow. Turing pattern formation without shear flow can occur in an activator-inhibitor system as long as the diffusivity of the inhibitor is larger than the diffusivity of the activator. In the presence of shear flow, a homogeneous steady state can become unstable even if this condition is not satisfied. Chemical patterns arise as a result of this instability. We study this instability in a simple system consisting of two layers moving relative to each other. We carry out a linear stability analysis showing the onset of the instability as a function of the relative speed between the layers. We solve numerically the nonlinear reaction-diffusion-advection equations to obtain these patterns. We find stationary, oscillatory, and drifting patterns extending along each layer. We also find regions of bistability that allow the formation of localized structures. The instability is analyzed in terms of Taylor dispersion.