What are three situations where Odysseus's guile is helpful to him?
First, once Odysseus realizes that Polyphemus isn't going to play by the rules of Greek hospitality, he speaks to the Cyclops with "words of guile," telling him that his ship was wrecked on the rocks on the beach and that only himself and the handful of men with him had escaped death. In telling such a clever lie, Odysseus prevents Polyphemus from going to look for his ship and killing the rest of his crew. Worse comes to worst, only he and the men he brought with him would die, and the rest would live.
Then, when Odysseus realizes that he's going to need to keep Polyphemus alive so that the Cyclops can remove the stone from the cave's mouth and let the men out, he devises a plan to blind him. Odysseus quite cleverly tells Polyphemus that his name is "Nobody," so that when other Cyclopes come to his aid after Odysseus's men blind him, Polyphemus tells them that "Nobody" is hurting him, and they leave. It's a pretty ingenious part of his plan.
Further, when Odysseus lands in Phaeacia and sees the young Nausicaa on the beach, he has enough guile to flatter her in order to win her favor and aid. Remember, he is naked and pretty worse for wear, having just endured Poseidon's awful storm, so when he approaches the princess, he needs to defuse the situation. He addresses her by asking, "'Are you some god or mortal?'" He goes on, talking about how much pleasure her parents and brothers must feel when they look at her, and how happy her future husband will be. In fact, he says that he's never seen a woman as beautiful as she. Now, I'm sure Nausicaa is very pretty, but Odysseus has lain with actual goddesses. He is intelligent and cunning enough to know just how to speak to this young maiden to get her to do exactly what he needs.
One situation is when he is trying to escape from Polyphemus's cave with his men; his plan to escape underneath the rams, and do so without detection, shows his cunning and guile. Another situation is in his dealings with Circe; he makes her promise to change his crew back into men and to use no more magic on him. Here his guile is useful in gaining the respect of Circe and securing a safe, relaxing respite in Circe's palace. Another example of his guile occurs in helping Telemachus deal with the suitors who have taken over Odysseus's palace. He uses his disguise to move among the suitors and see how they are treating his house and servants. In disguising himself, he is able to plan the most appropriate revenge to take upon the suitors.