The Flag of the United Federation of Planets
The Flag of the United Federation of Planets is a banner that consists of a blue background, with a white seal of the Federation above the words "UNITED FEDERATION of PLANETS."
On some occasions, the flag read "...OF PLANETS" with capital OF, such as DS9: "Take Me Out to the Holosuite."
In 2285, Kruge noted that with the Genesis Device, his officers could be sitting happily on their homeworld, with the exception of the flag of the Federation soaring above instead of that of the Klingon empire. (Star Trek III: The Search for Spock)
The Flag of the Federation is often draped over the photon torpedo caskets of fallen Starfleet officers. (DS9: "The Sound of Her Voice", "Tears of the Prophets")
In the 23rd century, Constitution-class starships' displayed it in their briefing rooms, such as the USS Defiant. (ENT: "In a Mirror, Darkly, Part II")
The flag is also present at Starfleet courts martial, such as the one for the Nova Squadron in 2368. (TNG: "The First Duty")
It was also displayed in Deep Space 9's briefing room during various official ceremonies, such as Worf's court martial, and the induction ceremony when Bajor (almost) joined the Federation. (DS9: "Rules of Engagement", "Rapture")
In accordance with an old Earth tradition, the flag was flown at a baseball game on the holosuite on Deep Space 9 when the Niners and the Logicians played a game in 2375. They also played the Anthem of the United Federation of Planets, taking their hats off and facing the flag. (DS9: "Take Me Out to the Holosuite")
The first "Flag of the Federation" in Star Trek was visible across Spocks coffin in Star Trek II: The Wrath of Khan. As it hadn't been designed yet, the Flag was missing from the Enterprise's conference room through out the Original Series.
As described in Star Trek: Action!, in Deep Space Nine "Tears of the Prophets", someone was looking at the photon torpedo prop and couldn't quite put their finger on what was missing. Until they disappeared, and came back with the Flag of the Federation to drape over it.