Well, to qualify as Supernatural, I'm guessing that your story has some paranormal or fantastic element (witches, demons, psychic powers, vampires, whatever).
Thriller and Mystery can be argued as different sub-genres, so it's probably not both. In a mystery, a central problem is discovered early on (a murder victim, a stolen diamond necklace), and the Protag's job is to discover who the bad guy is. In a thriller, your Protag may not know why he's being targeted by a maniacal killer, but he's running all the same.
Think of the difference between "24" (thriller) and "Matlock" (mystery).