We theoretically study the instability of helical shear flows, in which one fluid component flows along the vortex core of the other, in phase-separated two-component Bose-Einstein condensates at zero temperature. The helical shear flows are hydrodynamically classified into two regimes: (1) a helical vortex sheet, where the vorticity is localized on the cylindrical interface and the stability is described by an effective theory for ripple modes, and (2) a core-flow vortex with the vorticity distributed in the vicinity of the vortex core, where the instability phenomena are dominated only by the vortex-characteristic modes: Kelvin and varicose modes. The helical shear-flow instability shows remarkable competition among different types of instabilities in the crossover regime between the two regimes.
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