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The domestic cat is one of the most popular pets in the world. It is estimated that 89–92% of domestic cats in the UK are non-pedigree Domestic shorthair (DSH), Domestic longhair (DLH), or Domestic semi-longhair cats (DSLH). Despite their popularity, little is known of the UK non-pedigree cats’ population structure and breeding dynamics. Using a custom designed single nucleotide variant (SNV) array, this study investigated the population genetics of 1344 UK cats. Principal components analysis (PCA) and fastSTRUCTURE analysis verified that the UK’s DSH, DLH, and DSLH cats are random-bred, rather than admixed, mix breed, or crossbred. In contrast to pedigree cats, the linkage disequilibrium of these random-bred cats was least extensive and decayed rapidly. Homozygosity by descent (HBD) analysis showed the majority of non-pedigree cats had proportionally less of their genome in HBD segments compared to pedigree cats, and that these segments were older. Together, these findings suggest that the DSH, DLH, and DSLH cats should be considered as a population of random-bred cats rather than a crossbred or pedigree-admixed cat. Unexpectedly, 19% of random-bred cat genomes displayed a higher proportion of HBD segments associated with more recent inbreeding events. Therefore, while non-pedigree cats as a whole are genetically diverse, they are not impervious to inbreeding and its health risks.