The common potato (Solanum tuberosum L.) is an important staple crop with a highly heterozygous and complex tetraploid genome. The other taxa of cultivated potato contain varying ploidy levels (2X–5X), and structural variations are common in the genomes of these species, likely contributing to the diversification or agronomic traits during domestication. Increased understanding of the genomes and genomic variation will aid in the exploration of novel agronomic traits. Thus, sequencing data from twelve potato landraces, representing the four ploidy levels, were used to identify structural genomic variation compared to the two currently available reference genomes, a double monoploid potato genome and a diploid inbred clone of S. chacoense. The results of a copy number variation analysis showed that in the majority of the genomes, while the number of deletions is greater than the number of duplications, the number of duplicated genes is greater than the number of deleted ones. Specific regions in the twelve potato genomes have a high density of CNV events. Further, the auxin-induced SAUR genes (involved in abiotic stress), disease resistance genes and the 2-oxoglutarate/Fe(II)-dependent oxygenase superfamily proteins, among others, had increased copy numbers in these sequenced genomes relative to the references.