Linguistic universals arise from the interaction between the processes of language learning and language use. A test case for the relationship between these factors is linguistic variation, which tends to be conditioned on linguistic or sociolinguistic criteria. How can we explain the scarcity of unpredictable variation in natural language, and to what extent is this property of language a straightforward reflection of biases in statistical learning? We review three strands of experimental work exploring these questions, and introduce a Bayesian model of the learning and transmission of linguistic variation along with a closely matched artificial language learning experiment with adult participants. Our results show that while the biases of language learners can potentially play a role in shaping linguistic systems, the relationship between biases of learners and the structure of languages is not straightforward. Weak biases can have strong effects on language structure as they accumulate over repeated transmission. But the opposite can also be true: strong biases can have weak or no effects. Furthermore, the use of language during interaction can reshape linguistic systems. Combining data and insights from studies of learning, transmission and use is therefore essential if we are to understand how biases in statistical learning interact with language transmission and language use to shape the structural properties of language.This article is part of the themed issue ‘New frontiers for statistical learning in the cognitive sciences’.