The answer is none. Kim Davis has broken no law. The law in Kentucky says marriage is between a man and a woman. That law has not been changed by the Kentucky legislature (legislature being a body of elected officials tasked with making law, not to be confused with the judiciary). Also, it is not against the law to refuse to issue a marriage license no matter what reason you give for doing so. It certainly could be grounds for dismissal if that is one of your job requirements, but it is not illegal.
You might say “contempt of court” for failing to obey an injunction. Here’s how the Legal Information Institute at Cornell defines an injunction:
They are issued early in a lawsuit to maintain the status quo by preventing a defendant from becoming insolvent or to stop the defendant from continuing his or her allegedly harmful actions.
Harmful actions? Who has Kim Davis harmed by personally refusing to issue a marriage license to a same-sex couple? It goes on:
Injunctive relief is a discretionary power of the court in which the court, upon deciding that the plaintiff’s rights are being violated, balances the irreparability of injuries and inadequacy of damages if an injunction were not granted against the damages that granting an injunction would cause.
So apparently a particular county clerk in Kentucky can cause “irreparable injury” to someone by not issuing them a marriage license. Also, injunctions are only available to the court in cases of in-personam jurisdiction, meaning the court has authority over the person, as opposed to in-rem which means they have jurisdiction over the subject matter or particular location. Presumably Judge Bunning believes federal courts have authority over county employees regarding the performance of their job duties. By the way, this is not the first time Bunning has tried to use his office to bend local governments to his will (nor the first time he’s gone to bat for homosexual activists). In 2003 and again in 2006 he ordered the Boyd County Kentucky board of education to require students to undergo “diversity” training regarding sexual orientation.
Kim Davis is not legally obligated to issue marriage licenses to same-sex couples, or to anyone else. Issuing marriage licenses is one of her job duties. A judge has no more authority to hold her in contempt for not issuing marriage licenses than he does to hold in contempt a county employee who refuses to limit their breaks to 15 minutes twice a day (if that were a job requirement). Refusal to perform a job duty is not illegal and is not a matter for the courts – even for government employees. It’s certainly not a matter for federal courts when the job function in question is a state or local one.
Same-sex marriage is a fiction. The irreparable harm its proponents are concerned about is to the notion that two people of the same gender constitute a marriage. If dissent of any kind is allowed, others may realize the emperor has no cloths and they cannot have that. Kim Davis is in jail because she refused to bow the knee to the federal government. It’s as simple and as totalitarian as that. And that should frighten us all.