The effects of structural variants on phenotypic diversity and evolution are poorly understood. We recently described the genetic and phenotypic variation among fission yeast strains and showed that genome-wide association studies are informative in this model. Here we extend this work by systematically identifying structural variations and investigating their consequences. We establish a curated catalog of copy number variants (CNVs) and rearrangements, including inversions and translocations. We find that SVs frequently vary within clonal populations and are weakly tagged by SNPs, consistent with rapid turnover. We show that CNVs produce measurable effects on gene expression both within and outside the duplicated regions. CNVs contribute to quantitative traits such as cell shape, cell growth under diverse conditions, sugar utilization in winemaking and antibiotic resistance, whereas rearrangements are strongly associated with reproductive isolation. Collectively, these findings have broad implications for evolution and for our understanding of quantitative traits and complex human diseases.