What rights does my girlfriend have to my house after living there for 3 years?

Anyone can sue for anything, but that doesn't mean that she will win.  If she agrees with the facts as you have presented them, she would have no claim to any interest in the house.  Howver, if she claims that you she was a part owner of the house and that you were only holding title in your name as a convenience, she may make a claim that you were holding  her interest in the house in what is known as a "constructive trust", and the case may not be a slam dunk either way.  Normally, contracts involving interests in real property have to be in writing, but that is not always the case in dealings between fiduciaries, i.e. parent/child, attorney/client, and, depending on your exact circumstances, possibly live in boyfriend/girlfriend.  Frankly, I would consider romantic partners who lived together for more than 5 years to be fiduciaries.  This doesn't mean that you will lose, but it does mean that your ex, depending on exactly what she says the facts are, may have a viable claim (although I think your claim is stronger, particulary if you paid the entire down payment, and if you hadn't lived together before the house was purchased.)