Is there a DNA test to identify dog mixes?
There are many commercially available mixed breed ID tests, as a quick Google search will tell you.  However, I cannot vouch for their accuracy: this news article may be of interest. Also this.

The genetic markers used to identify breeds in a mixed-breed are microsatellite markers (sometimes SNPs): you can read more here, but from the abstract:


  We used molecular markers to study genetic relationships in a diverse collection of 85 domestic dog breeds. Differences among breeds accounted for ∼30% of genetic variation. Microsatellite genotypes were used to correctly assign 99% of individual dogs to breeds.


If you're interested in dog genetics, I encourage you to check out Elaine Ostrander's work.  She's written at least one book for laypeople, in addition to her numerous journal publications.