Heterogeneity in the environment favours foragers that are flexible (phenotypically plastic). However, consistent individual differences in behaviour (personality), such as in risk‐taking, might affect an individual's ability to find food, avoid predators and adjust to new conditions. It is now well known that personalities exist in many taxa, but much less is known about individual variation in plasticity. We measured the tendency to begin foraging across different levels of risk in house sparrows (Passer domesticus), using a behavioural reaction norm framework to simultaneously assess personality and plasticity. We asked whether individuals were consistently different across risk levels, and whether they differed in habituation and neophobia, both of which were treated as cases of plasticity. We found that males habituated more than females by beginning to feed sooner after repeated instances of a human disturbance in the presence of an initially unfamiliar object. Individuals of both sexes also exhibited consistent differences across trials, but did not differ in the rate of habituation beyond the difference between the sexes. When a novel object was placed in the foraging area, both sexes exhibited similar degrees of neophobia by delaying feeding. The magnitude of this change varied among birds, indicating individual differences in neophobia. Our results indicate that both personality and individual variation in plasticity exist but should be treated as independent phenomena. The presence of variation in plasticity implies that the raw material necessary for selection on neophobia exists, and that if heritable, plasticity in risk‐taking across contexts could evolve.