The linear stability of a fluid confined between two coaxial cylinders rotating independently and with axial sliding (spiral Couette flow) is examined. A wide range of experimental parameters has been explored, including two different radius ratios. Zeroth-order discontinuities are found in the critical surface; they are explained as a result of the competition between the centrifugal and shear instability mechanisms, which appears only in the co-rotating case, close to the rigid-body rotation region. In the counter-rotating case, the centrifugal instability is dominant. Due to the competition, the neutral stability curves develop islands of instability, which considerably lower the instability threshold. Specific and robust numerical methods to handle these geometrical complexities are developed. The results are in very good agreement with the experimental data available, and with previous computations.