Many theoretical and experimental studies suggest that range expansions can have severe consequences for the gene pool of the expanding population. Due to strongly enhanced genetic drift at the advancing frontier, neutral and weakly deleterious mutations can reach large frequencies in the newly colonized regions, as if they were surfing the front of the range expansion. These findings raise the question of how frequently beneficial mutations successfully surf at shifting range margins, thereby promoting adaptation towards a range-expansion phenotype. Here, we use individual-based simulations to study the surfing statistics of recurrent beneficial mutations on wave-like range expansions in linear habitats. We show that the rate of surfing depends on two strongly antagonistic factors, the probability of surfing given the spatial location of a novel mutation and the rate of occurrence of mutations at that location. The surfing probability strongly increases towards the tip of the wave. Novel mutations are unlikely to surf unless they enjoy a spatial head start compared to the bulk of the population. The needed head start is shown to be proportional to the inverse fitness of the mutant type, and only weakly dependent on the carrying capacity. The precise location dependence of surfing probabilities is derived from the non-extinction probability of a branching process within a moving field of growth rates. The second factor is the mutation occurrence which strongly decreases towards the tip of the wave. Thus, most successful mutations arise at an intermediate position in the front of the wave. We present an analytic theory for the tradeoff between these factors that allows to predict how frequently substitutions by beneficial mutations occur at invasion fronts. We find that small amounts of genetic drift increase the fixation rate of beneficial mutations at the advancing front, and thus could be important for adaptation during species invasions.