The Scottish SPCA is appealing for information after the crossbreed puppy was spotted in Duke Street and handed in to a police station.
Officers were so shocked by the animal's appearance they took her to a local vet clinic for emergency treatment and care.
The dog, who animal welfare workers have named Jess, weighed only 12 pounds – less than half her ideal weight.
Scottish SPCA senior inspector Billy Linton said: "Jess is probably the most emaciated living dog I've ever seen, she is effectively a walking skeleton.
"Weight loss on this level does not happen overnight and we believe she may have been neglected over a prolonged period, although we can't rule out the possibility that she may have been trapped or abandoned in a property.
"Her collar was hanging off her neck and she was severely anaemic and weak.
"Her feet and legs were heavily stained with urine and faeces, suggesting she has been kept in a kennel or cage for some time.
"Jess is still unwell. She received a blood transfusion on Monday after collapsing at the vet's and her gut is badly damaged due to the extent of her starvation.
"However, we are hopeful she will make a recovery."
Jess, a collie cross, is thought to be around a year old and was found by a member of the public last Friday.
Causing an animal unnecessary suffering is an offence and anyone found guilty of doing so can expect to be banned from owning animals for a fixed period or life.
Anyone with information relating to Jess is being urged to contact the Scottish SPCA's animal helpline on 03000 999 999.
Mr Linton added: "We think she is just a young dog and that she's around a year old, and a collie cross lurcher type.
"Despite everything she's been through she is still friendly and trusting of humans – which is both incredible and heart- breaking at the same time."