The genomes of hybrids often show substantial deviations from the features of the parent genomes, including genomic instabilities characterized by chromosomal rearrangements, gains, and losses. This plastic genomic architecture generates phenotypic diversity, potentially giving hybrids access to new ecological niches. It is however unclear if there are any generalizable patterns and predictability in the type and prevalence of genomic variation and instability across hybrids with different genetic and ecological backgrounds. Here, we analyzed the genomic architecture of 204 interspecific Saccharomyces yeast hybrids isolated from natural, industrial fermentation, clinical, and laboratory environments. Synchronous mapping to all eight putative parental species showed significant variation in read depth indicating frequent aneuploidy, affecting 44% of all hybrid genomes and particularly smaller chromosomes. Early generation hybrids with largely equal genomic content from both parent species were more likely to contain aneuploidies than introgressed genomes with an older hybridization history, which presumably stabilized the genome. Shared k-mer analysis showed that the degree of genomic diversity and variability varied among hybrids with different parent species. Interestingly, more genetically distant crosses produced more similar hybrid genomes, which may be a result of stronger negative epistasis at larger genomic divergence, putting constraints on hybridization outcomes. Mitochondrial genomes were typically inherited from the species also contributing the majority nuclear genome, but there were clear exceptions to this rule. Together, we find reliable genomic predictors of instability in hybrids, but also report interesting cross- and environment-specific idiosyncrasies. Our results are an important step in understanding the factors shaping divergent hybrid genomes and their role in adaptive evolution.