Well, you can always sue, but whether you can recover damages is another question. Usually you can sue neither the police/jurisdiction or the complainant if that person swears out a complaint that you were committing a crime. If you were totally exonerated of the charges, you may be able to sue them or the municipality for malicious prosecution, but that might in turn require showing they KNEW the charges were bogus and retaliatory. Unless you were incarcerated or injured during the arrest or suffered damage to your reputation that cost you business (i.e., newspaper stories), going after your tormentors is going to be difficult and costly, with little reward even if you win ($1 nominal damages, a "moral victory", maybe not worth spending thousands of dollars on if you don't have money to burn).
If you can show they knew their charges were false, in NY that is a misdemeanor crime of filing false reports and depending on what they said in court could also be actionable perjury, however, again, you have to show both the charges were false and they knew they were false. That's a tall order.
This answer is provided under the Avvo.com “Terms and Conditions of Use” (“ToU”), particularly ¶9 which states that any information provided is not intended as legal advice or to create an attorney-client relationship between you and me or any other attorney. Such information is intended for general informational purposes and should be used only as a starting point for addressing your legal issues. In particular, my answers and those of other attorneys on Avvo.com are not a substitute for an in-person or telephone consultation with an attorney licensed to practice in your jurisdiction about your specific legal issue. I can only answer your question based on the brief statement of facts you present in your question which may omit other relevant facts which would be disclosed in an in-person interview. While I attempt to provide reasonable care in answering your questions in an online forum which encourages attorneys to provide brief "off the cuff" responses within several hours of the posting of a question, I cannot perform any further detailed legal research concerning possibly relevant statutory or case law which might apply to your situation that I would normally do in the course of my representation of clients where an attorney-client relationship exists (and you are paying for my services!). You should therefore not rely solely upon legal information you obtain from this website or other resources which may be linked to an answer for informational purposes. You understand that questions and answers or other postings to the Site are not confidential and are not subject to attorney-client privilege. The full Avvo ToU are set forth at http://www.avvo.com/support/terms . In addition, while similar legal principles often apply in many states, I am only licensed to practice in the State of New York and Federal Courts. Any general information I provide about non-New York laws should be checked with an attorney licensed to practice in your State. Lastly, New York State Court rules (22 NYCRR Part 1200, Rule 7.1, available online at http://jackle.bo/_prof ) also require me to inform you that my answers and attorney profile posted on the Avvo.com site may be considered "attorney advertising" and that "prior results do not guarantee a similar outcome".
You can sue but you will probably have a hard time finding a lawyer to take the case on contingency so you will have to pay by the hour.
The above answer, and any follow up comments or emails is for informational purposes only and not meant as legal advice.
All calls, statements and reports to cops, all law enforcement, and all government boards, agencies and offices, as well as during legal/court proceedings and paperwork are protected and immune from civil lawsuits no matter how false, frequent or evil. So you can not sue your neighbor for this. Making such false reports is a crime, but only the government can prosecute crimes, you can not. You can do your best to bring the falsity to the authorities' attention, and if they want, they can investigate and perhaps prosecute, but you can not force them.
If you can prove your neighbor's trespass, call the cops.
We do not have an attorney-client relationship. I am not your lawyer. The statements I have made do not constitute legal advice. Any statements I have made are based upon the very limited facts you have presented, and under the premise that you will consult with a local attorney. This is not an attempt to solicit business. This disclaimer is in addition to any disclaimers that this website has made. I am only licensed in California.