Copy number variants (CNVs) are a pervasive source of genetic variation and evolutionary potential, but the dynamics and diversity of CNVs within evolving populations remain unclear. Long-term evolution experiments in chemostats provide an ideal system for studying the molecular processes underlying CNV formation and the temporal dynamics with which they are generated, selected, and maintained. Here, we developed a fluorescent CNV reporter to detect de novo gene amplifications and deletions in individual cells. We used the CNV reporter in Saccharomyces cerevisiae to study CNV formation at the GAP1 locus, which encodes the general amino acid permease, in different nutrient-limited chemostat conditions. We find that under strong selection, GAP1 CNVs are repeatedly generated and selected during the early stages of adaptive evolution, resulting in predictable dynamics. Molecular characterization of CNV-containing lineages shows that the CNV reporter detects different classes of CNVs, including aneuploidies, nonreciprocal translocations, tandem duplications, and complex CNVs. Despite GAP1's proximity to repeat sequences that facilitate intrachromosomal recombination, breakpoint analysis revealed that short inverted repeat sequences mediate formation of at least 50% of GAP1 CNVs. Inverted repeat sequences are also found at breakpoints at the DUR3 locus, where CNVs are selected in urea-limited chemostats. Analysis of 28 CNV breakpoints indicates that inverted repeats are typically 8 nucleotides in length and separated by 40 bases. The features of these CNVs are consistent with origin-dependent inverted-repeat amplification (ODIRA), suggesting that replication-based mechanisms of CNV formation may be a common source of gene amplification. We combined the CNV reporter with barcode lineage tracking and found that 102-104 independent CNV-containing lineages initially compete within populations, resulting in extreme clonal interference. However, only a small number (18-21) of CNV lineages ever constitute more than 1% of the CNV subpopulation, and as selection progresses, the diversity of CNV lineages declines. Our study introduces a novel means of studying CNVs in heterogeneous cell populations and provides insight into their dynamics, diversity, and formation mechanisms in the context of adaptive evolution.