Effective surveillance is critical to managing biological invasions via early detection and eradication. The efficiency of surveillance systems may be affected by the spatial arrangement of sample locations. We investigate how the spatial arrangement of sample points, ranging from random to fixed grid arrangements, affects the probability of detecting a target population (survey sensitivity) and the overall cost of detecting and eradicating populations invading over time. For single period surveys, regular sampling patterns outperform the equivalent number of random samples at intermediate sample densities, but only when sample sensitivity is high. Otherwise, sample point arrangement has little effect on survey sensitivity, which can be modelled reasonably accurately using a Poisson approximation. For multiple period surveys, we find little difference in the costs of sample point arrangements for most combinations of parameters tested. However, the costs of different arrangements vary when sampling methods have higher sensitivity and trap densities are low, a situation representative of many real surveillance programs. In particular, our results suggest that dynamic trapping arrangements increase the efficiency of detection when traps are sparse relative to the size of target populations. Also, for the scenarios we considered managers may exercise some freedom in allocating sample point locations. Placing individual traps or samples in perceived higher probability sites at the local scale is unlikely to diminish the probability of detection at the broader scale.