Swan Song is the phrase that members of the Black Swan use before they think a situation will make them unable to carry on their usual work as a member. In human terms, the swan song is a metaphorical phrase for a final gesture, effort, or performance given just before death or retirement. When one calls their swan song, it means that they are warning other Black Swan members of impending danger.
We know three members who have called their swan song: Prentice Endal, right before he was apprehended by the Council (though nobody knows how he knew so early); Mr. Forkle, right before he passed on in the remnants of Lumenaria in Lodestar; and Jolie Ruewen, before she died in a fire. Keefe Sencen has also used the phrase during Everblaze as a code word, but it was not a true swan song.
The phrase "swan song" seems like it can also be called when Black Swan members are stranded on a mission, and they can call for help by calling it. An example of this is seen in Neverseen, when Sophie Foster, Keefe Sencen, and Mr. Forkle went on a mission to interrogate Gethen, Mr. Forkle told them to say "swan song" should they meet with any difficulties. This alternative usage is different from the main usage and likely tends to be used more often by Black Swan members.
It can also be used as a code or a password, as seen when in Everblaze, Jolie Ruewen gives the phrase "swan song" to her spectral mirror, Vertina, and tells her that only those who speaks these two words to her can be trusted with the secret that she took to the grave (for information on the secret, see Jolie Ruewen).