JK Rowling discussed this in an interview in 2006 (in a question about Muggles brewing potions):
I been asked what would happen if a Muggle picked up a magic wand in my world. And the answer would probably be something accidental... possibly quite violent. Because a wand, in my world, is merely a vehicle — a vessel for what lies inside the person.
Squibs might have residual magic passed through their family tree, and there was a question about non-human creatures using wands recently (goblins, house-elves, etc.) who all have an innate magical ability, but Muggles generally have none. (Or so little that it can’t be put to effect.)
In the 2007 Bloomsbury web chat, JK Rowling was discussing how Muggle-borns acquire their ability, and suggested that it was a partially inherited trait:
How exactly do Muggle-borns receive magical ability?
Muggle-borns will have a witch or wizard somewhere on their family tree, in some cases many, many generations back. The gene re-surfaces in some unexpected places.
So Muggles may have a very small part of the magical gene buried inside them, but this is probably so small as to be a rounding error. This means that there’s effectively nothing for the wand to channel, so they couldn’t use it to perform magic.
This also explains the potential variation in reaction from the wand: a Muggle with absolutely no magical blood might get nothing from a wand, whereas somebody with a long-forgotten magical ancestor might get some sparks or a small explosion.
However, there doesn’t seem to be any law in canon which specifically bans a Muggle from possessing a wand. But somebody who gave a wand to a Muggle who was unaware of the magical world (whether deliberately or by accident) might be in breach of the Statute of Secrecy.