Theoretical framework for linear stability of an anomalous sub-diffusive activator-inhibitor system is set. Generalized Turing instability conditions are found to depend onanomaly exponents of various species. In addition to monotonous instability, known fromnormal diffusion, in an anomalous system oscillatory modes emerge. For equal anomaly exponents for both species the type of unstable modes is determined by the ratio of the reactants'diffusion coefficients. When the ratio exceeds its normal critical value, the monotonous modesbecome stable, whereas oscillatory instability persists until the anomalous critical value isalso exceeded. An exact formula for the anomalous critical value is obtained. It is shownthat in the short wave limit the growth rate is a power law of the wave number. Whenthe anomaly exponents differ, disturbance modes are governed by power laws of the distinctexponents. If the difference between the diffusion anomaly exponents is small, the splittingof the power law exponents is absent at the leading order and emerges only as a next-ordereffect. In the short wave limit the onset of instability is governed by the anomaly exponents,whereas the ratio of diffusion coefficients influences the complex growth rates. When theexponent of the inhibitor is greater than that of the activator, the system is always unstabledue to the inhibitor enhanced diffusion relatively to the activator. If the exponent of theactivator is greater, the system is always stable. Existence of oscillatory unstable modes isalso possible for waves of moderate length.