What does Odysseus learn from the Sirens?
In the twelfth book of Homer's Odyssey, Odysseus tells the people of Phaeacia about his journey from Circe's island of Aeaea. Before he leaves, Circe gives him some advice about the journey ahead. One of the obstacles in his path are the Sirens, who are part-female and part-bird. If a person is not prepared for the encounter, Circe says, then "Whoever encounters them unawares and listens to their voices will never joy at reaching home" (A.S. Kline translation). Circe also tells Odysseus how he can manage to listen to the Sirens' song without losing his life.
At Odyssey 12.165-200, Odysseus plugs up his crews' ears with wax and has them tie him to the mast of their ship. As Odysseus approaches, the Sirens sing the following song:
Famous Odysseus, great glory of Achaea, draw near, and bring your ship to rest, and listen to our voices. No man rows past this isle in his dark ship without hearing the honeysweet sound from our lips. He delights in it and goes his way a wiser man. We know all the suffering the Argives and the Trojans endured, by the gods’ will, on the wide plains of Troy. We know everything that comes to pass on the fertile Earth. (A.S. Kline translation)
From the Sirens' song, it appears that Odysseus will have gained some wisdom, although this wisdom does not seem to "kick in" until Odysseus makes it back to Ithaca, because not long after his encounter with the Sirens, he falls asleep on Thrinacia and his men kill the cattle of the Sun.
Still, Odysseus is wise enough to survive his encounter with the Sirens, so this does add to his own personal glory.
I suppose that it is also possible that the Sirens could have given Odysseus some glimpse at past events of which he was unaware or what the future would hold, given that they "know everything that comes to pass." Homer, however, is silent about what precisely Odysseus might have learned from them.
This is a puzzling encounter, though, and this type of riddling bird-woman can also be seen in Oedipus' encounter with the Sphinx.