A rotating fluid flow between differentially translating parallel plates, which induce uniform suction and injection, is studied as a canonical model of swirling flow where suction, shear and Coriolis effects compete. This relatively simple modelling yields several reduced equations that are valid for asymptotically large suction, shear and/or rotation rates. The linear stability problems derived from the full Navier–Stokes and reduced problems are numerically solved and compared. In addition to Taylor-vortex modes, transverse-roll-type instabilities are found in Rayleigh-stable and -unstable parameter regions when weak suction is applied. These instabilities, separated by the so-called Rayleigh line, are characterised by vortices attached to the suction wall. Another type of instability, which exists beyond the Rayleigh line and shows inviscid motion in the fluid core, is found when suction is sufficiently strong. The relation of this instability to the stability results by Gallet, Doering & Spiegel (Phys. Fluids, vol. 22, 2010, 034105) is discussed. Our nonlinear analyses indicate subcritical and supercritical bifurcations of finite-amplitude solutions for the near-wall and fluid-core instabilities, respectively.