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According to a study by Paws With a Cause, fewer than one shelter dog in a hundred is capable of becoming a service dog. In their study they found that only one shelter dog in four was even adoptable to start with. Some were reclaimed by owners, while some were ill or temperamentally unsuited to be pets. Only 6.5% of the shelter animals they temperament tested were considered acceptable for service work and were taken for further screening. This 6.5% of shelter dogs had improved chances of adoption even if they did not make it through training to become service dogs because of the work done by the program in evaluating them. That 6.5% weren't necessarily saved but were definitely helped.
Seventy-five percent of the dogs taken for further testing wash out due to hip or elbow dysplasia which wouldn't put them out of the running for a good pet home but would make it cruel to the dog to place them in a working home.
Of the 1.5% of dogs who meet both temperament and health requirements, only 1 in 8 is actually able to complete the training. Those that wash out during the training phase can easily be rehomed into pet homes because the training they do receive makes them excellent well-mannered companions. The waiting list for these dogs is usually years long.
One in 500 makes it from shelter to a job as a service dog, but for each dog that makes it as a service dog seven more are also saved. Animals who make it part way through the program are better trained and more adoptable because of participating in the program. The program also takes an active role in finding the dog a forever home.