The rule that solves most cases
Ask who issued or certified the document, not who needs it or where that person happens to live. In the United States, each of the 50 states, the District of Columbia and the territories has its own apostille authority — almost always the Secretary of State or an equivalent office — and that authority only apostilles documents issued, recorded or notarized within its own jurisdiction. The federal government, through the U.S. Department of State's Office of Authentications, apostilles only documents that come from a federal agency, a federal court or a federal official acting in that capacity.
Put differently: the question is not "which state do I live in?" or "which country am I sending this to?" It is "where does the signature or seal that needs authenticating come from?" A birth certificate, a marriage certificate, a state court ruling or a notarial act almost always belongs on the state route. A document signed by a federal official — such as an FBI background check — follows the federal route, regardless of which state the requester happens to live in.
Example: living in Florida does not turn a New York birth certificate into a Florida document. That certificate has to be apostilled through New York, because New York recorded the birth and only New York can certify the signature of the official who issued it.