The first omen we encounter is in book 2, between likes 145 and 178. The gathered men look on with astonishment, and Halistherses steps forwards to interpret the omen, explaining that Odysseus will soon return and disaster will fall on the suitors.
Book 15 contains an eagle-omen too (around line 160), where an eagle flies past Telemachos as he prepares to leave Sparta. This bird is carrying a goose in its talons; Helen interprets that Odysseus will swoop down on those who have become so comfortable and well-fed living in his house, just as the goose had been in the palace yard.
In book 15, again, at line 526ish a bird flies past Telemachos on his right - this time, rather than an eagle, this is a hawk, one of the signifying birds of Apollo. This carries a dove, and is tearing and scattering its feathers around the ship. This one is not interpreted quite as clearly as the previous two, but Theoklymenos states that it signifies that Telemachos’ family shall hold sovereign power on Ithaka forever.
At book 20 line 241 we see another eagle holding a dove - this one flies in on the left side of the suitor Amphinomos, and he quickly interprets that their plan to kill Telemachos is not going to be successful.
