Previous evolutionary analyses of human culture have found that a simple model of random copying, analogous to neutral genetic drift, can generate the distinct power-law frequency distribution of cultural traits that is typical of various real-world cultural datasets, such as first names, patent citations and prehistoric pottery types. Here, we use agent-based simulations to explore the effects of frequency-dependent copying (e.g., conformity and anti-conformity) on this power-law distribution. We find that when traits are actively selected on the basis of their frequency, then the power-law distribution is severely disrupted. Conformity generates a ‘winner-takes-all’ distribution in which popular traits dominate, while anti-conformity generates a ‘humped’ distribution in which traits of intermediate frequency are favoured. However, a more passive frequency-dependent ``trimming’’, in which traits are selectively ignored on the basis of their frequency, generates reasonable approximations to the power-law distribution. This frequency-dependent trimming may therefore be difficult to distinguish from genuine random copying using population-level data alone. Implications for the study of both human and nonhuman culture are discussed.