Structural variants have a considerable impact on human genomic diversity. However, their evolutionary history remains mostly unexplored. Here, we developed a new method to identify potentially adaptive structural variants based on a network-based analysis that incorporates genotype frequency data from 26 populations simultaneously. Using this method, we analyzed 57,629 structural variants and identified 577 structural variants that show high population distribution. We further showed that 39 and 20 of these putatively adaptive structural variants overlap with coding sequences or are significantly associated with GWAS traits, respectively. Closer inspection of the haplotypic variation associated with these putatively adaptive and functional structural variants reveals deviations from neutral expectations due to (i) population differentiation of rapidly evolving multi-allelic variants, (ii) incomplete sweeps, and (iii) recent population-specific negative selection. Overall, our study provides new methodological insights, documents hundreds of putatively adaptive variants, and introduces evolutionary models that may better explain the complex evolution of structural variants.