When a neutral mutation arises in an invading population, it quickly either dies out or ‘surfs’, i.e. it comes to occupy almost all the habitat available at its time of origin. Beneficial mutations can also surf, as can deleterious mutations over finite time spans. We develop descriptive statistical models that quantify the relationship between the probability that a mutation will surf and demographic parameters for a cellular automaton model of surfing. We also provide a simple analytic model that performs well at predicting the probability of surfing for neutral and beneficial mutations in one dimension. The results suggest that factors – possibly including even abiotic factors – that promote invasion success may also increase the probability of surfing and associated adaptive genetic change, conditioned on such success.