The idea seems to have been that you'd immortalize yourself or your own ugly baby (a face that only a mother could love!) or your dog. Of course, things turned out differently.
During a trial run by Stamps.com last year, The Smoking Gun managed to order stamps with the likenesses of Ted Kaczynski, the Rosenbergs, Slobodan Milosevic, Jimmy Hoffa, and others from the Book of Infamy. You can find these here. The Smoking Gun's point was that "Nobody's dog deserves to be on a stamp."
After the trial, the USPS authorized three private companies to print "custom stamps":
I was very attracted by the idea of getting postage with subversive messages made and decided to give Zazzle a try. Herewith are some my custom stamps that were rejected. The images are also available here. All of the images are from art by Doug Minkler. You can find an extensive collection of digital images of his art on his website.