In principle, it would be possible to take two blood samples at different times from the same animal, extract and sequence the DNA then compare the sequences, but it would be very unlikely that you could detect a spontaneous mutation that way because these events are rare and sequencing DNA is not 100% accurate. However, it is possible to do something like this with cancer tumours when a mutation has caused a change in the DNA that radically alters the cells behaviour.
What we do most often is compare the DNA from many different individuals and compare them to a reference genome that has been sequenced and annotated as accurately as possible. Many of the changes occur in just one nucleotide base of a DNA molecule: These are called SNPs (Single Nucleotide Polymorphisms) and some change the gene product resulting from transcription and translation of the DNA sequence. These are called ‘functional’ SNPs. Other functional SNPs outside the ‘coding’ region of a gene have no direct impact on the gene-product, but often affect the control of gene expression.
Even if we don’t know what effect a SNP has in the genome, these features are very useful as ‘biomarkers’ that we can use to determine the ancestry of an individual by studying the ‘crossing-over’ of DNA strands when single-stranded DNA from the sperm and egg of two mating individuals unite during fertilisation. This crossing-over is a major source of genetic diversity from which natural selection can favour individuals with genes that make them fit to survive in a given environment, such as the harsh conditions in the remote Scottish Mountains where Wildcats live.