A moon dog (moondog), or paraselene, is a lunar halo caused by light refraction off ice crystals high in the atmosphere. A link can be found below.
Well there is a "Moon Dog" [See related question] and the "Moon Rover" \:) However, as to an actual dog, the answer has to be no - not unless one of the astronaut's managed to sneak one onto one of...
Shorings bank (New York)
Only one - Pluto.
There hasn't been a dog on the moon yet.