The stability of circular Poiseuille–Couette flow to axisymmetric disturbances is investigated theoretically. First, the governing circular Orr–Sommerfeld equation for linear perturbations is formulated and analysed asymptotically at large values of the Reynolds number. The existence of multiple regions of instability is predicted and their dependence upon radius ratio and inner cylinder velocity is determined explicitly. These findings are confirmed when the linear problem is solved numerically at finite Reynolds number and multiple neutral curves are found. The relevance of these results to the thread injection of medical implants is discussed, and it is shown how the linear modes are connected to nonlinear amplitude-dependent modes at high Reynolds number that exist for $O(1)$ values of the inner cylinder velocity.