The linear stage of stability is studied for a two-component fractional reaction-diffusion system. It is shown that, with a certain value of the fractional derivative index, a different type of instability occurs. The linear stability analysis shows that the system becomes unstable toward perturbations of finite wave number. As a result, inhomogeneous oscillations with this wave number become unstable and lead to nonlinear oscillations which result in spatial oscillatory structure formation. A computer simulation of a Bonhoeffer-van der Pol type of reaction-diffusion system with fractional time derivatives is performed.