What are some allusions in Books 10-12 of Homer's The Odyssey?
Odysseus alludes to the Trojan War when he sees his mother in the Underworld in Book X. He tells her that he's been wandering and experiencing "endless hardship from that day [he] first set sail with King Agamemnon bound for Troy, the stallion-land, to fight the Trojans there." The Trojan War was, indeed, a terrible ordeal, discussed at length in The Iliad. Odysseus fought in the Trojan War for ten years, prior to the start of his odyssey home, and the Achaeans only, finally, breached the walls of Troy and defeated with Trojans with the deceptive Trojan Horse (Odysseus's idea).
Odysseus sees many important figures from mythology in the Underworld. He describes the wives and daughters and mothers of quite a few famous men. He says, "And I saw Alcmena next, Amphitryon’s wife, who slept in the clasp of Zeus and merged in love and brought forth Heracles, rugged will and lion heart." With this line, he alludes to the story of Hercules. Zeus came to Hercules's mother in the form of her husband, so she slept with him, and she got pregnant with Hercules, son of Zeus and a demigod.
Moreover, Odysseus continues, "And I saw Megara too, magnanimous Creon’s daughter wed to the stalwart Heracles, the hero never daunted." Megara's story is terribly sad: Hera, in her anger at Zeus for his affair with Alcmena, Hercules's mother, drove Hercules insane so that he killed his wife, Megara, and their children. Hercules, when she returned him to his senses, felt so terrible that he embarked on a mission to cleanse himself of his awful actions.
Further, Odysseus sees "the mother of Oedipus, beautiful Epicaste. What a monstrous thing she did, in all innocence—she married her own son ... who’d killed his father, then he married her!" Another awful story: Epicaste receives a prophecy that her son would grow up to kill his father and marry her, so when she gave birth to Oedipus, she sent him away to be killed. However, the slave who took him away didn't kill him; he gave the baby away to a shepherd who served another royal family. When Oedipus grows up, he leaves home and an oracle tells him the same prophecy, so he vows never to go home again (not knowing he was adopted). In trying to avoid the prophecy, he inadvertently makes choices that allow for it to come true: he kills his real father and marries his mother, even siring children by her. When Epicaste finds out, she kills herself.
Books 10-12 of Homer's Odyssey contain a number of allusions to other events in classical mythology.
In Book 10 and elsewhere, Hermes is given the epithet "Slayer of Argus," which alludes to a story told outside of Homer about how Hermes killed a multi-eyed creature named Argus, who was charged with guarding a woman named Io.
Book 11 contains dozens of allusions as Odysseus sees or encounters the spirits of numerous famous dead people. For example, Odysseus sees "Eriphyle, who sold her own husband’s life for gold." (A.S. Kline translation). This is an allusion to Amphiaraus of Argos, whose wife Eriphyle, bribed by Polyneices with a divinely-made necklace, convinced her husband to go off and fight in a war that he knew would result in his own (Amphiaraus') death.
In the same book, Odysseus sees Heracles, who mentions that he "served a man far inferior to me, and he set me difficult tasks." This is an allusion to Eurystheus, the king who sent Heracles on his famous labors.
Finally, in Odyssey 12, we find an allusion to the quest for the Golden Fleece when Circe tells Odysseus about the Wandering Rocks:
Only one ocean-going vessel has passed between them, the celebrated Argo fleeing from Aeetes, and the waves would have quickly broken her on the massive crags, if Hera had not seen her through, because of her care for Jason.