The epidemiology and transmission dynamics of sheep scrapie is as yet poorly understood. Here we present a theoretical analysis of the transmission dynamics within a sheep flock, concentrating on how persistence properties depend on transmission scenario and flock size. Patterns of disease persistence and extinction are studied analytically using branching-process approximations and numerically using stochastic model simulations. For a given basic reproduction number, disease extinction is most likely when late-stage infected animals are responsible for most of the transmission. This effect can be understood in terms of aggregation in the distribution of the number of secondary infections arising from a single primary infection. The presence of an environmental reservoir reduces the probability of extinction.