We study the stability of plane Poiseuille flow (PPF) and plane Couette flow (PCF) subject to streamwise system rotation using linear stability analysis and direct numerical simulations. The linear stability analysis reveals two asymptotic regimes depending on the non-dimensional rotation rate : a low- and a high- regime. In the low- regime, the critical Reynolds number and critical streamwise wavenumber are proportional to, while the critical spanwise wavenumber is constant. In the high- regime, as, we find and for streamwise-rotating PPF, and and for streamwise-rotating PCF, with. Our results for streamwise-rotating PPF match previous findings by Masuda et al. (J. Fluid Mech., vol. 603, 2008, pp. 189-206). Interestingly, the critical values of and at in streamwise-rotating PPF and PCF coincide with the minimum reported by Lezius & Johnston (J. Fluid Mech., vol. 77, 1976, pp. 153-176) and Wall & Nagata (J. Fluid Mech., vol. 564, 2006, pp. 25-55) for spanwise-rotating PPF at and PCF at. We explain this similarity through an analysis of the perturbation equations. Consequently, the linear stability of streamwise-rotating PCF at large is closely related to that of spanwise-rotating PCF and Rayleigh-Bénard convection, with, where is the critical Rayleigh number. To explore the potential for subcritical transitions, direct numerical simulations were performed. At low, a subcritical transition regime emerges, characterised by large-scale turbulent-laminar patterns in streamwise-rotating PPF and PCF. However, at higher, subcritical transitions do not occur and the flow relaminarises for. Furthermore, we identify a narrow range where turbulent-laminar patterns develop under supercritical conditions.