We study the nonlinear mode competition of various spiral instabilities in magnetised Taylor–Couette flow. The resulting finite-amplitude mixed-mode solution branches are tracked using the annular-parallelogram periodic domain approach developed by Deguchi & Altmeyer (Phys. Rev. E, vol. 87, 2013, 043017). Mode competition phenomena are studied in both the anticyclonic and cyclonic Rayleigh-stable regimes. In the anticyclonic sub-rotation regime, with the inner cylinder rotating faster than the outer, Hollerbach et al. (Phys. Rev. Lett., vol. 104, 2010, 044502) found competing axisymmetric and non-axisymmetric magneto-rotational linearly unstable modes within the parameter range where experimental investigation is feasible. Here we confirm the existence of mode competition and compute the nonlinear mixed-mode solutions that result from it. In the cyclonic super-rotating regime, with the inner cylinder rotating slower than the outer, Deguchi (Phys. Rev. E, vol. 95, 2017, 021102) recently found a non-axisymmetric purely hydrodynamic linear instability that coexists with the non-axisymmetric magneto-rotational instability discovered a little earlier by Rüdiger et al. (Phys. Fluids, vol. 28, 2016, 014105). We show that nonlinear interactions of these instabilities give rise to rich pattern-formation phenomena leading to drastic angular momentum transport enhancement/reduction.