In this paper we investigate stability of travelling wave solutions to a class of reaction-diffusion equations perturbed by infinite-dimensional additive noise with Hölder continuous paths, covering in particular fractional Brownian motion with general Hurst index. We obtain long- and short time asymptotic error bounds on the maximal distance from the solution of the stochastic reaction-diffusion equation to the orbit of travelling wave fronts. These bounds, in terms of Hurst index and Hölder exponent, apply to a large class of infinite-dimensional self-similar drivers with Hölder continuous paths, such as linear fractional stable motion. We find that for short times, higher Hurst indices imply higher stability, while for large times, a smaller gap between Hurst index and Hölder exponent implies stability for larger noise amplitudes.