We examined 11 non-linear regression models to determine which of them best fitted curvilinear species accumulation curves based on pit-trapping data for reptiles in a range of heterogeneous and homogenous sites in mesic, semi-arid and arid regions of Western Australia. A well-defined plateau in a species accumulation curve is required for any of the models accurately to estimate species richness. Two different measures of effort (pit-trapping days and number of individuals caught) were used to determine if the measure of effort influenced the choice of the best model(s). We used species accumulation curves to predict species richness, determined the trapping effort required to catch a nominated percentage (e.g. 95%) of the predicted number of species in an area, and examined the relationship between species accumulation curves with diversity and rarity. Species richness, diversity and the proportion of rare species in a community influenced the shape of species accumulation curves. The Beta-P model provided the best overall fit (highest r2) for heterogeneous and homogeneous sites. For heterogeneous sites, Hill, Rational, Clench, Exponential and Weibull models were the next best. For homogeneous habitats, Hill, Weibull and Chapman–Richards were the next best models. There was very little difference between Beta-P and Hill models in fitting the data to accumulation curves, although the Hill model generally over-estimated species richness. Most models worked equally well for both measures of trapping effort. Because the number of individuals caught was influenced by both pit-trapping effort and the abundance of individuals, both measures of effort must be considered if species accumulation curves are to be used as a planning tool. Trapping effort to catch a nominated percentage of the total predicted species in homogeneous and heterogeneous habitats varied among sites, but even for only 75% of the predicted number of species it was generally much higher than the typical effort currently being used for terrestrial vertebrate fauna surveys in Australia. It was not possible to provide a general indication of the effort required to predict species richness for a site, or to capture a nominated proportion of species at a site, because species accumulation curves are heavily influenced by the characteristics of particular sites.