I am a big fan of adult dogs from shelters- they are easier to care for at that age and you know exactly what you are getting. I am more cautious about pups from shelters and rescues. If you are flexible about the ultimate size, shape and temperament of you dog, then a shelter puppy should work out great. I have seen "rottweiler-mix" puppies that grew into smallish adult dogs and a "cocker spaniel-mix" puppy that weighs 10 pounds and looks like a chi-jack cross, so don't count on the pup turning into the breed the shelter pegs them for.
My other worry with shelter pups is parvo and distemper. I have seen several pups from the shelters who got sick shortly after being adopted, sticking their new owners with hefty vet bills. The odds of a shelter puppy getting sick are significantly higher than for a puppy from a good breeder, so this is a serious concern. This is certainly something to think about when adopting pups from inside the shelter (if they are housed in foster care this is not such a risk) that have had limited vaccines. You may want to talk to your vet about this as risk varies quite a bit. Vets should know of any recent outbreaks in your area that you should be concerned about.