Compared to human genetics, the canine genetics field is in its infancy. However, the canine genome was sequenced in 2005 and is increasingly well annotated with genes and variants. As well as conventional Sanger sequencing being possible, there are high density chip arrays which allow simultaneous assay of hundreds of thousands of single base pair variations (SNPs) across the genome and these make performing gene mapping studies possible.
The unique way in which dogs have been bred to perform different jobs and look in certain ways has led to dogs being particularly valuable as a way to investigate the genetic basis of disease. Firstly, within breeds there is often less genetic diversity than would be expected in a human population of similar size. That can lead to enrichment for a gene or small number of genes for particular traits – good (such as desirable conformation or behaviour) or bad (predisposing to disease). That makes tracking down what those genes are easier. Secondly, it means dogs have an unusual genetic haplotype structure that makes gene mapping uniquely tractable in this species. Whereas in humans, thousands of cases and controls are needed to map disease associations on a genome-wide basis, in dogs simple traits can be mapped with as few as 20 individuals in case and control groups and complex traits with as few as 100 in each group.
There is a good review about gene mapping in dogs here.