Adaptation from standing genetic variation or recurrent de novo mutation in large populations should commonly generate soft rather than hard selective sweeps. In contrast to a hard selective sweep, in which a single adaptive haplotype rises to high population frequency, in a soft selective sweep multiple adaptive haplotypes sweep through the population simultaneously, producing distinct patterns of genetic variation in the vicinity of the adaptive site. Current statistical methods were expressly designed to detect hard sweeps and most lack power to detect soft sweeps. This is particularly unfortunate for the study of adaptation in species such as Drosophila melanogaster, where all three confirmed cases of recent adaptation resulted in soft selective sweeps and where there is evidence that the effective population size relevant for recent and strong adaptation is large enough to generate soft sweeps even when adaptation requires mutation at a specific single site at a locus. Here, we develop a statistical test based on a measure of haplotype homozygosity (H12) that is capable of detecting both hard and soft sweeps with similar power. We use H12 to identify multiple genomic regions that have undergone recent and strong adaptation in a large population sample of fully sequenced Drosophila melanogaster strains from the Drosophila Genetic Reference Panel (DGRP). Visual inspection of the top 50 candidates reveals that in all cases multiple haplotypes are present at high frequencies, consistent with signatures of soft sweeps. We further develop a second haplotype homozygosity statistic (H2/H1) that, in combination with H12, is capable of differentiating hard from soft sweeps. Surprisingly, we find that the H12 and H2/H1 values for all top 50 peaks are much more easily generated by soft rather than hard sweeps. We discuss the implications of these results for the study of adaptation in Drosophila and in species with large census population sizes.