Dogs do have bones in their tails. Dog tails are essentially the last section of the dog's spine. A dog's tail can include as many as 23 vertebrae. Shorter natural full tails can have as few as six vertebrae.
Some dogs are born with no tail or a tail that is naturally very short and stubby. Some breeds have a docked tail as part of the breed standard. These are often herding or working dogs, for which a long tail can be dangerous for the dog and those around it. Tails are connected to versatile muscles with tendons and ligaments that give the dog precise control over the entire tail.