You can't stop people from filing police reports, even if they are false. It's up to the police to investigate the complaint and take the appropriate action. If someone lied out of hand, then the police may charge the person with obstructing a police investigation but again, this is the police's call, not yours. The only option you have is suing someone for defamation of character if the statements made against you were completely untrue.
You can't press charges for false accusations, but you may be able to sue in civil court the person who made the untrue statements.
If someone made untrue statements about you to the police and these statements caused you material harm, then you may be able to sue for defamation of character. Defamation is a catch-all term covering two different types of lawsuits – "libel" for written defamation and "slander" for spoken defamation. Filing a false police report could be either, or both, depending how the accusation was made. Defamation is not a crime and you can't press charges for it. Rather, you would sue the person who made the untrue statements in a civil court.
Police Reports May be Privileged
Generally, false police reports are protected from defamation claims by the law of privilege. This means the person cannot be held liable for the statements he made even if they could be considered defamatory. The reasoning here is that citizens should be encouraged to report potential criminals to the police without the threat of legal action if they get the facts wrong. In most states, privilege applies as long as the person filed the report in good faith. If she did it just to annoy or harass you, then you may be able to argue that privilege does not apply. The law is complicated, so speak to a lawyer about your options.
Proving Material Harm
One of the tests for defamation is material or "cognizable" harm. This means you must suffer substantial damage that is generally measurable in terms of dollars. For example, if you were fired from your job as a direct result of someone's false accusations against you, that would be a cognizable harm. Defamation law does not recognize emotional injury in most cases. It's unlikely that you could file a defamation claim if all you suffered was temporary anxiety or the inconvenience of having the police question you.
Proving the Untrue Statements
In a civil defamation case, the burden is on you to prove that the accusations made against you were false. Truth is an absolute defense to defamation so if there are facts to back up the person's statement, you are not entitled to compensation – it's not enough to show that the person was wrong about some of the details. You must have clear and convincing evidence that what was said about you was categorically untrue, for example, documents, emails, timelines and witness testimony.