We investigated how an invading organism’s dispersal characteristics affect the efficacy of different surveillance strategies aimed at detecting that organism as it spreads following a new incursion. Specifically, we assessed whether, out of the surveillance strategies tested, the best surveillance strategy for an organism varied depending on the way it disperses. We simulated the spread of invasive organisms with different dispersal characteristics including leptokurtic and non-leptokurtic kernels with different median dispersal distances and degrees of kurtosis. We evaluated surveillance strategies with different sampling arrangements, densities and frequencies. Surveillance outcomes compared included the time to detection, the total spread of the invasion and the likelihood of the invasion reaching new areas. Overall, dispersal characteristics affected the surveillance outcomes, but the grid surveillance arrangement consistently performed best in terms of early detection and reduced spread within and between fields. Additionally, the results suggest that dispersal characteristics may influence spread to new areas and surveillance strategies. Therefore, knowledge on an invasive organism’s dispersal characteristics may influence how we search for it and how we manage the invasion to prevent spread to new areas.