Where do I file my LLC operating agreement?
Nowhere. In most instances, you will not file your LLC Operating Agreement with anyone or anything.
You are not required to file your LLC Operating Agreement with a state agency, at least as a requirement for having a validly formed and fully-operational LLC under applicable state law. In most instances, you will keep your LLC Operating Agreement to yourself as a private contract among you and your fellow business owners (if applicable). However, please do not read this and then fall into the trap of the unwise–thinking that if something is not required to be filed with a court or government agency, it must not be important enough to worry about. If we used that line of thinking, almost no business contracts, trusts, powers of attorney or a host of other legal documents would exist–as most of such key legal papers are likewise never filed with a court or other government agency. If you and I think carefully, we come to realize that there are a host of reasons why we write and rely on written agreements, other than just because certain documents must be filed or submitted to the court or government.
To repeat, you most likely are not required to file your LLC Operating Agreement anywhere or with any person or entity–at least insofar as you are simply in the process of forming and ensuring proper administration of your LLC. Even so, you will want to commission a qualified attorney to help you draft a customized Operating Agreement that will serve as the basis for ownership and operating of your business now and into the future.
P.S. as usual, when we are talking about Limited Liability Company matters, most of the same the concepts and principles noted above regarding LLCs and operating agreements are equally applicable for corporations and their related bylaws and detailed articles of incorporation. Likewise, partnerships and partnership agreements can also be generally analogized to these things.
 Of course, there could be exceptions–there are always exceptions, right? If this were a law school exercise in Socratic method or otherwise a discussion among attorneys, we could go into a list of such potential exceptions and instances where you might want or be required to file the LLC Operating Agreement with a court (i.e. in the case of pending civil litigation or perhaps a probate), with a county recorder or with some other government agency or civil official. But if you are asking the original question, most of the time you are asking because you have just formed or are thinking about forming your new LLC, have heard a thing or two about Operating Agreements and want to know about the requirements associated with the same, including where such legal document will need to filed.
 Self-serving comment as an attorney who drafts such legal papers? Yes. But true nonetheless.