How has evolution led to the variation in behavioural phenotypes (personalities) in a population? Knowledge of whether personality is heritable, and to what degree it is influenced by the social environment, is crucial to understanding its evolutionary significance, yet few estimates are available from natural populations. We tracked three behavioural traits during different life-history stages in a pedigreed population of wild house sparrows. Using a quantitative genetic approach, we demonstrated heritability in adult exploration, and in nestling activity after accounting for fixed effects, but not in adult boldness. We did not detect maternal effects on any traits, but we did detect a social brood effect on nestling activity. Boldness, exploration and nestling activity in this population did not form a behavioural syndrome, suggesting that selection could act independently on these behavioural traits in this species, although we found no consistent support for phenotypic selection on these traits. Our work shows that repeatable behaviours can vary in their heritability, and that social context influences personality traits. Future efforts could separate whether personality traits differ in heritability because they have served specific functional roles in the evolution of the phenotype, or because our concept of personality and the stability of behaviour needs to be revised.