A piece of legislation by the British Ministry of Magic. Clause 3 states that no non-human creature is allowed to possess a wand. (GF9)
When Winky was found with a wand following the unrest in the World Cup campground (GF9), wizarding authorities assumed she stole it and used it to conjure the Dark Mark. Besides all this, her very possession of a wand was a crime under clause 3 of the Code of Wand Use and almost certainly contributed to her sacking.
The goblins participating in the Chipping Clodbury Riots (DP) were in direct violation of clause 3 of the Code of Wand Use.
Many non-human creatures (such as house-elves, goblins, and centaurs) are capable of performing their own kinds of magic without wands. Possession of a wand seems to be seen as symbolic of power and dignity equal to that of wizards and witches. Thus, the prohibition on wand ownership for non-humans represents systemic institutionalized inequality and fear of an uprising. Note that the use of wands throughout history seems to have given witches more equal status with their male counterparts than in seen among Muggles and that ministry policy under the Thicknesse/Death Eater regime held that Muggle-borns could not rightfully possess wands, either. -BB
This way of thinking about wands lends itself well to a Freudian reading. -BB