In the state of Kentucky how can a minor become emancipated?

First, you have to be at least 16. You file a petition with the court. Your parents/guardian receive a copy of the petition and a summons to appear at the hearing. Once the hearing is held, the court can issue an emancipation order if it's determined that emancipation is in your best interests and you have demonstrated the ability to manage your financial affairs, including proof of employment or other means of support (can not be public assistance) and have provided proof of housing. Even if the court grants the order, your parents can appeal it. And the emancipation order can be rescinded if your circumstances change (ie you become homeless or lose your means of support)

That's a very brief summary of the process--there's actually much more involved than that.