While she never carried a flag or a flag officer, Enterprise is always referred to as a the "flagship of the Federation", at least by her own crew and some aliens of dubious importance. This is despite never having been part of any task force, battle group, or fleet.
Being the flagship of the Federation has its drawbacks. The flagship of the Federation's speed is limited to warp factor one and it is curiously vulnerable to being disabled by Klingon ships that are called "birds-of-prey" because an early draft of Star Trek II involved Romulans (even though the film instead featured Klingons), that are about 1/10 of flagship of the Federation's size (In Search of Spock), or outright destroyed by a single photon torpedo. (Star Trek: That TNG Movie Starring William Shatner)