This paper investigates the problem of recovering the support of structured signals via adaptive compressive sensing. We examine several classes of structured support sets, and characterize the fundamental limits of accurately recovering such sets through compressive measurements, while simultaneously providing adaptive support recovery protocols that perform near optimally for these classes. We show that by adaptively designing the sensing matrix we can attain significant performance gains over non-adaptive protocols. These gains arise from the fact that adaptive sensing can: (i) better mitigate the effects of noise, and (ii) better capitalize on the structure of the support sets.