General:
* Canard (a false or unfounded report or story)
* Misinformation
* Disinformation (intentionally false information spread to deceive)
* Fabrication
* Lie
* Untruth
* Gossip
* Scandal (if it's damaging to someone's reputation)
More specific:
* Slander (false and defamatory statement)
* Libel (written or published false and defamatory statement)
* Hoax (a deliberate deception intended to trick or mislead)
* Urban legend (a widely-circulated story, often with a supernatural element, believed to be true but usually false)
The best synonym for "false rumor" will depend on the context of your writing.