Although personality has been defined as a suite of correlated behaviours, most studies of animal personality actually consider correlations between a few traits. We examined the repeatability and correlational structure of five potential personality traits (activity, neophobia, exploratory tendencies, risk-taking behaviour and obstinacy), in female zebra finches. In addition, we assessed to what extent personality influenced social dominance in a feeding context in this gregarious species. All personality traits were found to be highly repeatable within individuals. In addition, except for obstinacy, all of them were related to each other, thus defining a behavioural syndrome. Social dominance was predicted by personality, with proactive individuals being more likely to be dominant. Our results suggest that personality can be considered as a new static factor influencing within-group hierarchies. We finally discuss these results in terms of the consequences for the evolution of personalities and the need to take several traits into account to provide full descriptions of individual personality.