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.
Depending on how you view Kurumi's feelings towards Shido, she is either Cute and Psycho or Yandere. Or both. Or even a Tsundere, as a growing number of fans believe.
The fact that in Date A Live: Rinne Utopia, one of her clones (the one that did fall for Shido) was proven to be a Yandere in one of her bad endings, and in her situational drama CD she (and Shido) displayed Tsundere characteristics, served to support interpretations that weren't plain Cute and Psycho.
Even haters of the anime (or even the series in general) love her, or at least regard her to be the one saving grace of the anime. People who hate the cliché Harem premise (or hate the anime for any other reason, usually being its extremely poor start) have said that they're enduring the first few anime episodes until they get to see her in action (and she didn't disappoint). And there are those who dropped the anime until she showed up.
Evil Is Cool: Inverse Form Tohka is much more dangerous than her base form, and has so far displayed the greatest destructive force of all Spirits. It says something when she obliterates the majority of the belligerents and nearly levels the city during Volume 7's climax (end of second season). With just one swing.
Compared to the rest of the spirits, who are chaste and innocent, Kurumi is openly seductive and quite frank and comfortable with her sexuality. She knows the effectiveness of her eroticism on men, and not shy about using it to her advantage.
Ellen Mira Mathers. Between the sensual design of her CR suit and her Sexy Secretary looks, topped by her insatiable lust for battle, she gets to rival all of the above with unrelenting gusto. Wescott is such a lucky bastard.
Friendly Fandoms: There is a good chance that a Date A Live fan is also a High School D×D fan. It also helps the fact that both series are from the same publisher.
Holy Shit Quotient: Beyond the silly Excuse Plot is a story that deals with apocalyptic beings, as well as the potential to harvest their powers for nefarious ends, so when the series takes a dark turn, expect fans to scream this.
I Am Not Shazam: A weird case here; the title of Volume 7 has led some fans to think that the Spirit on the cover is Miku and that something happened to her, when it's actually Tohka's Superpowered Evil Side. Also inverted by other fans who recognise the Spirit, but get the title wrong as a result.
Jerkass Woobie: Miku and Natsumi. The former is Broken Bird who pretty much lost all hope in humanity before Shido came along, and the latter is extremely self-deprecating, taking on a fake appearance just to receive what she would perceive as praise.
Les Yay: Miku aside, there's loads of this between Origami and Mikie in the AST. Covered in one of the spinoff manga, no less.
Magnificent Bastard: Special mention goes to Sir Isaac Ray Peram Westcott. Developing a gambit to throw Tohka over the edge of the Despair Event Horizon and invoke her inverse form that he's been trying to harness after thinking up several ways to get her to do so, including Cold-Blooded Torture... That should give you an idea that he's probably becoming one of these.
Memetic Mutation: Mikimoto saying "Mahal kita." (Tagalog for "I love you.") People posting comments on forums talk about it a lot.
Any of Shido's deaths are this either when he gets shot with a huge hole on his body, or the fact that he's all covered in blood after crossing a five meter barrier that would freeze him if any Mana was detected.
Holding up your hand, realizing that it's not there because it got cut by a laser blade. Thank you for that, Westcott and Ellen.
One of Rinne Utopia's bad ending has Shidou visiting Rinne's house on a whim, the player has a choice to ring the house's intercom or not, if the player choose to ring it, Shidou will enter her house and discover that not only the house was empty but it is also AN EXACT REPLICA of Shidou's house, in fear Shidou starts realizing that he doesn't remember anything at all aboout Rinne, all of sudden Ruler appears and warned him before the screen turns black, presumably killing Shidou.
In Volume 3, just the fact that Kurumi devours her offenders, along with some of the animal-abusers that were attacking an innocent cat, is enough to creep you out. Also, there's Shido's appropriate reaction to the bloody mess that Kurumi had made when she ate the abusers.
Rescued from the Scrappy Heap: The series is really good at doing this. A single volume/episode can cause the fans to go from hating a girl to loving her. Case in point: Kotori, Miku, both of which started as unlikeable, but Character Development skyrocketed their popularity back to an acceptable level, if not more. It remains to be seen whether Origami will receive the same treatment.
Origami, because of her Stalker with a Crush tendencies, her constant attempts to ruin romantic moments (out of jealousy and her persistent prejudice against spirits), and the fact that she's really weak compared to the other girls. It doesn't help that she frequently makes things worse instead of better, such as interfering with Shido's attempts to tame Spirits and end up provoking them. The rest of the AST also get this treatment for the same reason, albeit to a smaller extent.
Kotori used to be this due to the her really annoying Dominatrix commander mode (and the fact that people were starting to get tired of Shana Clones). Then episodes 9-10 elevated her to Ensemble Darkhorse status.
Ai, Mai and Mii — but especially Mii. Their only purpose so far has been an unfunny Running Gag where they interpret Shido's (and sometimes other people's) actions in the worst way possible, and fans find Mii's constant "gross" — her only line — to be incredibly grating and unfair to Shido.
Stoic Woobie: Origami. The girl's parents were killed by a Spirit, yet she dutifully carries on as an AST soldier, silently waiting until she gets the chance to get back at the murderer. Then she finds out that she was the Spirit.