Background: The science of complex systems has been proposed as a way of understanding health services and the demand for them, but there is little quantitative evidence to support this. We analysed patterns of healthcare use in different urgent care settings to see if they showed two characteristic statistical features of complex systems: heavy-tailed distributions (including the inverse power law) and generative burst patterns. Methods: We conducted three linked studies. In study 1 we analysed the distribution of number of contacts per patient with an urgent care service in two settings: emergency department (ED) and primary care out-of-hours (PCOOH) services. We hypothesised that these distributions should be heavy-tailed (inverse power law or log-normal) in keeping with typical complex systems. In study 2 we analysed the distribution of bursts of contact with urgent care services by individuals: correlated bursts of activity occur in complex systems and represent a mechanism by which overall heavy-tailed distributions arise. In study 3 we replicated the approach of study 1 using data systematically identified from published sources. Results: Study 1 involved data from a PCOOH service in Scotland (725,000) adults, 1.1 million contacts) and an ED in New Zealand (60,000 adults, 98,000 contacts). The total number of contacts per individual in each dataset was statistically indistinguishable from an inverse power law (p > 0.05) above 4 contacts for the PCOOH data and 3 contacts for the ED data. Study 2 found the distribution of contact bursts closely followed a heavy-tailed distribution (p < 0.008), indicating the presence of correlated bursts. Study 3 identified data from 17 studies across 8 countries and found distributions similar to study 1 in all of them. Conclusions: Urgent healthcare use displays characteristic statistical features of large complex systems. These studies provide strong quantitative evidence that healthcare services behave as complex systems and have important implications for urgent care. Interventions to manage demand must address drivers for consultation across the whole system: focusing on only the highest users (in the tail of the distribution) will have limited impact on efficiency. Bursts of attendance - and ways to shorten them - represent promising targets for managing demand.