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Adaptation in extended populations often occurs through multiple independent mutations responding in parallel to a common selection pressure. As the mutations spread concurrently through the population, they leave behind characteristic patterns of polymorphism near selected loci—so-called soft sweeps—which remain visible after adaptation is complete. These patterns are well-understood in two limits of the spreading dynamics of beneficial mutations: the panmictic case with complete absence of spatial structure, and spreading via short-ranged or diffusive dispersal events, which tessellates space into distinct compact regions each descended from a unique mutation. However, spreading behaviour in most natural populations is not exclusively panmictic or diffusive, but incorporates both short-range and long-range dispersal events. Here, we characterize the spatial patterns of soft sweeps driven by dispersal events whose jump distances are broadly distributed, using lattice-based simulations and scaling arguments. We find that mutant clones adopt a distinctive structure consisting of compact cores surrounded by fragmented “haloes” which mingle with haloes from other clones. As long-range dispersal becomes more prominent, the progression from diffusive to panmictic behaviour is marked by two transitions separating regimes with differing relative sizes of halo to core. We analyze the implications of the core-halo structure for the statistics of soft sweep detection in small genomic samples from the population, and find opposing effects of long-range dispersal on the expected diversity in global samples compared to local samples from geographic subregions of the range. We also discuss consequences of the standing genetic variation induced by the soft sweep on future adaptation and mixing. [ABSTRACT FROM AUTHOR]