These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
The Romans were of two minds about Odysseus (or Ulysses, as they called him) - on one hand they claimed to be descended from Aeneas and his Trojans and thus saw him as an enemy, on the other hand they also were proud because in some versions of the myth they also were descended from him through his son or grandson Latinus, Aeneas's father-in-law.
Big Lipped Alligator Moment: The story of Ares and Aphrodite's love told by Demodocus. It's quite long and irrelevant. Many think this part is an interpolation.
Crowning Moment of Heartwarming: when Odysseus and Penelope finally get back together, they can't stop talking to each other and Athena has to personally delay the dawn goddess to give them time to talk, maybe do other things, and finally get a bit of sleep.
Designated Hero: There are many occasions when it's hard to see Odysseus as a hero.
Fridge Brilliance: Odysseus has been away from Ithaca for twenty years, and Telemachos is just beginning to take control of his family in the last six months or so. So who's been ruling Ithaca? That's right, Penelope.
Ho Yay: Eurylochus to Odysseus: "You're a hard man, Odysseus. Your fighting spirit's stronger than ours; your stamina never fails. You must be made of iron head to foot." Also, Telemachus and everyone.
Idiot Plot: but it seems that a good deal of Odysseus's problems either come from his own stupidity, or at the very least his crew's. But he's already been stated to be a master tactician, and he worships frickin' Athena, patron goddess of Smart Guys. It's like every time they land on an island, Odysseus gets pegged in the face with the idiot ball.
There is a reason behind this. The Greek strongly believed in a concept called ὕβρις, which is often rendered today as "hubris" and can be translated to modern days "Pride" or "Acting as a human shouldn't". Another thing that hubris carries along is punishment (if you are guilty of hubris, you are going to be punished somehow). All of Odysseus's mistakes are made out of hubris. For a better description of what hubris means, see The Other Wiki's explanation.
Narm: In one of the live action adaptations, Odysseus' mother being Driven to Suicide involves a shouting match between her and Penelope and a servant rolling around on the ground doing...something. It's very hard to take the scene seriously with all that.
Penelope: (blocking Odysseus' mom's path as she's about to walk into the ocean) RAGGHHHHHHHHHHHH!!!!!
Tear Jerker: The whole story of Odysseus' dog, Argos. Don't know the story? Well, Argos was his dog, who he trained when he was a pup. Then Odysseus had to leave for 20 years, going to Troy, trying to get back, ect. During his leave, the dog, since his master wasn't there anymore, had to live outside, staying on a pile of dung to keep warm, getting too old and sick to move anywhere else anyway. Odysseus eventually gets home and walks pass his dog (disguised as a beggar by Athena). Argos senses that the person is his master and proceeds to die by the happiness and shock of seeing his master again. And the worst part? Since Odysseus is pretending not to be himself, he can't even grieve for his now dead dog. You can't say that you didn't get a bit teary eyed when you read about that.
Also a bit of Writers Cannot Do Math, given that a dog is extremely unlikely to live that long in an era without veterinarians.
Then again, considering the genre, it could simply be Homer's way of telling us just how strong us the bond was; it's doubtful he and his audience were unaware of the relative shortness of an average dog's lifespan.
Odysseus seeing his mother's ghost (when he hadn't even known she was dead) and trying in vain to hold her is pretty sad, too.
Odysseus meeting his old friends and allies in Hades (especially when Ajax still refuses to speak with him even when Odysseus pleads with him to let bygones be bygones). In other news, Achilles is still a whiner.
Telemachus trying, and failing, to get the Ithacan assembly to condemn the suitors is a minor one.
Values Dissonance: What's the very first thing Odysseus does on his way home? He and his men make a halt on a foreign coast, where they attack and plunder a town, killing the men and taking the women as slaves. It's described as a completely normal thing for them to do.
Also, nowadays the killing of the handmaidens who slept with the suitors and the goatherd who allied himself with them seems rather...unnecessary, and are cut from most retellings. The ones that do leave it in (like the TV miniseries) usually cut it down to a single handmaiden who is unintentionally killed in the crossfire trying to aid the suitors.
Odysseus sleeping with several women and thus cheating on his wife during his journey home while Penelope remains steadfastly faithful would be considered to be rather jerkass behavior of Odysseus, but in Ancient Greece this behavior was rather normal.
What an Idiot: Odysseus is wise enough to listen to the advice he's given. His companions, on the other hand, never learn to take his direst warnings seriously... which is pretty much the reason why Odysseus comes home alone.
Odysseus also suffers from this in places. Like when he and his men have successfully escaped the Cyclops, he turns around and starts insulting him. Fair enough, except he also told the Cyclops his name, his father's name, and where he lives. This is a bit like slapping a gangster in the face and then giving him your wallet.
Also, Aeolus's bag of winds. If keeping it closed was so important that he didn't dare go to sleep while guarding it, why did he even need to bring it with him at all? Letting Aeolus hang onto the bag, opening it only after news of Odysseus's safe return reached him, would've saved Odysseus years of travel later and a whole lot of sleeplessness right away.
And Elpenor. Most of Odysseus' men who die do so for generally sound reasons (devoured by monsters, killed in a divine storm etc). How does Elpenor die? He goes to bed on the roof drunk, wakes up with a hangover, and proceeds to forget he's on the roof so falls off it and breaks his neck.What.
The Woobie: Odysseus, Penelope, and Telemachus, all in their own ways.