His treatment of Loki brings up the question if he ever did love him at all. Imprisoning him is one thing, disowning him is another. However, Loki did cause the senseless deaths of who knows how many people by this point, and didn't show so much as a hint of remorse or repentance. Is Odin just accepting that his son, who is now a war criminal, is too far gone and doesn't want his old feelings for him to outweigh his duty as king? Keep in mind that Loki became insane after falling off the Bifrost Bridge and could really use some therapy.
His stance on how to handle Malekith. Was he not thinking clearly due to Frigga's death, or was he showing himself Not So Different to the likes of Malekith and Loki? Was he resigned to fighting the Dark Elves at Asgard and let it be destroyed because he just didn't care anymore, or was he accepting the heavy cost that was the best chance at preserving the universe? Maybe he was absolutely right and was just voicing it poorly. Would a single ship of Dark Elves really be capable of defeating the Asgardians in Asgard when their entire race couldn't do so on their home planet? Furthermore, the Asgardians just had to hold the Dark Elves off for less than a day and the Convergence would have been over and it wouldn't matter anymore. Odin isn't wrong, he's just too upset to properly explain himself. He could be going senile. He's been trying to pass the throne for several years at this point, and he's growing increasingly erratic even before Malekith attacks, so he may even be aware that he's slipping.
Usurping the throne. Did he merely use his presumed death to his advantage to catch Odin off-guard; or did he manipulate everyone from Kurse to Thor in a plot to usurp the throne, intentionally setting up Frigga's death knowing Thor and Odin would react to it in the way that they did? Or was it somewhere in between: did he manipulate events expecting it would be Odin who died defending the Aether rather than Frigga, but still go through with his planned usurpation once she was avenged?
Saving Jane. Did he finally change for the better and save the love of his brother's life? Was he doing it to honor his late mother's sacrifice in protecting Jane? Or did he just want to make sure Thor leaves Asgard and joins Jane on Midgard instead?
Loki-as-Odin allowing Thor to pursue his own agenda as guardian of the Nine Realms. Did he do it out of good will as much as convenience, did getting Odin out of the picture temporarily sate his need for vengeance, or did Thor hand him a major gift (a way to get him out of the picture without being seen as dicking over the savior of the Nine Realms) without knowing it? Another option could be that Loki-as-Odin is trying not to blow his cover (among other things, had Thor tried to hand him Mjolnir, he would not have been able to hold it, thus immediately blowing his cover), so everything he said to Thor in that scene is what Odin would have said - i.e. real!Odin is wise enough to understand what his son is saying about being unfit to handle The Chains of Commanding etc. Another possibility is that his chat with Thor on the flying boat ("I wish I could trust you") really opened his eyes, and he knows that if Thor returns to Asgard with things as-is, everyone's going to prison or "the axe". So at least a partial motive (or rationalization) could be his intention to do his reconciled brother a good turn. Considering that he pardoned people (i.e., Thor's friends) who had offended him constantly throughout his life...
The film's finale: too much comedy or a good blending of action and humor?
The focus on Thor/Jane and Loki over Malekith and Asgard as a whole- either you ship the pairing and/or love Loki so this film's great for you, or you don't like the two (See below for reasons) and you think Loki's only being shoehorned in because of Hiddleson's popularity with the fans, and thus you don't really like a lot of the film.
Contested Sequel: Fans are heavily divided between whether this or the first one is the better Thor film, in contrast to other MCU series where there's usually one obvious favorite entry.
Fan-Preferred Couple: Some fans still pull for Thor and Sif over Thor and Jane, and in this case, they are supported by Odin.
Genius Bonus: The gag of Thor hanging up on his hammer on a coat hook in Jane's apartment makes perfect sense considering that typical warrior's etiquette when entering the personal home of a trusted friend is to hang up your weapons as a courtesy.
The meme below about Tom Hiddleston's infamous 'embrace' pose? Something like this happens in the actual movie when Thor winds up on a subway and a girl accidentally leans against him.
There's a theater in Shanghai using a fan-made poster of Loki photoshopped in Thor's arms in place of Jane as promotion.
Ho Yay: The interactions between Malekith and Algrim suggests something more than the latter having loyalty to the former. The scenes where Malekith gives Algrim the Kursed stone and when Algrim wants Malekith to rest after his battle in Asgard help.
Jerkass Has a Point: Nice as it is to see the star-crossed lovers, uh... cross the stars and get together after all, Odin and Loki have a point: his relationship with Jane is eventually going to expose Thor to heartbreak, in a mere 70 years or less. No matter which way you cut it, the relationship with Jane is doomed to fail, whether Thor likes it or not.
Jerkass Woobie: Weirdly enough, Loki falls into this once again. He's truly devastated when he hears that Frigga is dead, becoming little more than a broken mess until he gets the chance for revenge.
In the first few minutes of the film, he massacres many of his own men by dropping their ships on the enemies so that he and a skeleton crew, as well as his right hand man, can escape, then has the gall to blame Asgard for doing this.
When he first attacks Asgard, he causes many more deaths, including that of Frigga for trying to hide Jane Foster away from him. This last action is so reprehensible that Loki, of all people, is outraged and agrees to join up for revenge.
In the last act of the film, he tries destroying reality, something that he had attempted millenia before.
Nightmare Fuel: The black hole grenades. The screams of agony, the horrifying cracking noises and what they imply definitely paints a gruesome picture. The one that eventually kills Kurse is particularly slow and gruesome and at one point you can see it dragging his eyeballs out of their sockets!
Erik Selvig was liked just fine in the first movie and The Avengers, but that was mainly because he could be serious in addition to comical. Here he devolves into almost pure comic relief territory that many felt unnecessary and annoying. Helping matters is Selvig receiving a far smaller amount of screentime in comparison to the first movie, only approaching his original characterisation at the end of the film, but even that is undermined by the film still having Selvig act like a fool. Worse still, Selvig is repeatedly stated to have suffered a nervous breakdown after what Loki did to him, yet none of his fragile mental health is treated seriously and is entirely Played for Laughs. Thankfully, he's back to normal by Avengers: Age of Ultron.
Darcy's new intern/boyfriend gets some of this as well due to being rather flat, and even Darcy herself for those that don't consider her an Ensemble Darkhorse.
Excluding Betty Ross, Jane Foster possibly has the least amount of fans for a heroic female character in the MCU for her unbelievable romance with Thor that brought down the story and negatively affects the quality of the film, and also flattened her own character, keeping her entertainingly pathological obsession with her research from the last movie to a minimum in favor of her relationship drama. The fact that a lot of other interesting characters and plot points from the comic was short-changed in exchange for her did not help. Also not helping her cause is that, to some fans, Jane is simply not that interesting as a character, particularly in this movie. As said above, this could have been completely avoided by giving her love of science an emphasis, but Selvigg is given nearly all of the scientific importance instead, leaving Jane with almost nothing.
Loki takes a special delight in exchanging "pleasantries" with Sif, maintains lingering eye contact with her, and even calls her "ravishing" at one point (though he says this to Thor in Sif's form, not Sif herself). Of course, this is Loki we're talking about here, who loves to get under people's skin. It could also be considered a nod to the myths where those two were said to have had an affair.
After threatening to "pay her a visit" in Thor, Loki finally meets Jane. It goes about as well as expected at first - she slaps him for New York, and he says he likes her. But by the end of the film, he shields her from harm with his own body. Twice.
Ship-to-Ship Combat: There has been some Thor/Jane vs. Thor/Sif combat going on. Sif's actress, Jaimie Alexander, stirs the pot in an interview about Thor and Sif's relationship as well as scenes of the two acting way more than Just Friends probably contributed to this, though she also pointed out that Sif doesn't really have any negative feelings toward Jane or her relationship with Thor. A glare Sif gives Jane that many mistook for jealousy was actually because Asgard was threatened by Jane's presence. As Alexander points out, Sif is a warrior for her realm first and foremost.
Signature Scene: Loki masquerading as Captain America, played by Chris Evans.
Strangled by the Red String: Thor/Jane again. They knew each other for two days, maybe three, then pined for each other for two years even though they barely knew each other. Some critics believe there is little chemistry between the actors. The most convincing moment the couple have is their big kiss during the finale. However, rather tellingly, this wasn't even Natalie Portman but Chris Hemsworth's real-life partner acting as a stand in.
They Wasted a Perfectly Good Character: General reception to the movie is that while it's a fun action movie and has good set up for Guardians of the Galaxy, if you liked the supporting characters from the last film you'll be disappointed.
Sif, mainly because her actress was injured during filming and couldn't be in as much of the movie as she could have been otherwise.
The Warriors Three, but especially Hogun. Poor guy basically just stayed home and sat the whole adventure out.
Malekith never manages to be anything more than a Generic Doomsday Villain. What makes it especially painful is that he was played by Christopher Eccleston (who passed on the Doctor Who 50th anniversary special for this, despite reviving the role) and he's a much more entertaining character in the comics. Most people blame Loki receiving extra focus due to his popularity for this.
Frigga, who has little to no defining characteristics in the first film beyond "Odin's wife" and "Thor and Loki's mother". Just as they begin to expand her character in this one, she's promptly killed off, and her death is basically a plot point to get Thor and Loki to work together.
They Wasted a Perfectly Good Plot: We don't see much of Jane's reactions to visiting Asgard, and neither Darcy or Erik ask her what it was like, or observe how the experience could have changed her.
Jane when affected by the Aether (Giving her Black Eyes of Evil and, in one case, black eyes and blue pupils) looks wrong. Thank God we never see them in a close up or in detail.
The Dark Elves in their creepy, doll-like masks.
The Untwist: Everyone, both in- and out-of-universe, expects Loki to betray Thor. And sure enough, it looks like he does...except it's part of a ploy to get Malekith to draw the Aether out for Thor to destroy it. Ultimately played straight, as he fakes his death and usurps the Throne of Asgard while Thor is on Earth.
Visual Effects of Awesome: A staple of the Marvel Cinematic Universe, though the standout is the combination of practical and digital effects used to bring Kurse to life. As an example, we get to see more of Asgard this time around, and it is stunning. It really looks like something out of a Nordic haven, with lasers.
What an Idiot: Thor's entire plan after the initial Dark Elf attack. Bust Loki out of prison? Not a great idea. But the worse idea was to take Jane away from the heavily fortified Asgard, and leave behind all his allies who could be trusted to boot. Predictably, it ends poorly, with Malekith getting the Aether, and Loki usurping Odin's throne.