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.
Kara becoming a Red Lantern in the New 52. While being Hot-Blooded is a long established character trait, and she has plenty of reasons to be angry, many feel its just out of character to make her so Darker and Edgier by making her a member of a group known for being the most volatile of the Lantern groups. However, Charles Soule's handling of the story in his Red Lanterns book has been mostly well received, and many feel he made the story actually work.
The miniskirt costume. Some find it demeaning and stupid, considering the obvious implications of flying in a skirt and how regularly its used for fanservice. Others however (not including the ones who just like it for the fanservice) like the skirt because its so synonymous with her character, like Superman's trunks; its a stupid design choice, but something so iconic to her design at this point. The New 52 outfit is generally given a similar mixed reception, since it doesn't do anything to reduce the fanservice while also doesn't resemble her classic outfit at all.
Complete Monster: The Post-Crisis version of Benjamin Martin Krull, alias Reactron, was never a nice guy. His first clash with Supergirl saw a vast amount of property damage and significant casualties, with many survivors suffering from radiation sickness or cancer. It was in the New Krypton storyline, however, where Reactron really crossed the line. Having been rebuilt by Lex Luthor into a cyborg with a Gold Kryptonite heart, Reactron invades New Krypton alongside Metallo, depowering a killing numerous Kryptonians, threatening to rape Supergirl, and then murdering her father, Zor-El, in front of her by giving him radiation sickness. Tasked by General Lane with eliminating Supergirl, Flamebird, and Nightwing, Reactron murdered all the soldiers assigned to help him when they expressed doubts about the mission, and did his best to slowly torture Supergirl and Flamebird to death, all while gloating about how fighting two such attractive girls was a dream come true. Captured and tortured by Supergirl's mother, Alura, Reactron waited until Supergirl came to free him from the torture chamber to reveal that his capture had been a set up and that Luthor had transformed him into a living bomb. Detonating himself with a smile, Reactron gives a serene look as he incinerates Alura, the city of Kandor, and the entire planet of New Krypton, killing over ninety percent of the population in a single fell swoop. A lech, a bully, and a thug at his core, Reactron is a study in just how horrific it can be when somebody like that gets their hands on real power, and has done more to hurt Supergirl than any of her other adversaries, killing her father, her mother, and her entire race in a matter of weeks.
Fanon Discontinuity: Fans of Linda Danvers will refuse to accept how the character was treated in Reign in Hell and tend to ignore it completely.
Fridge Brilliance: The Pocket Universe "Matrix" Supergirl's being a protoplasmic artificial lifeform created by Lex Luthor makes a lot more sense considering how Lex's Silver Age origin involved him creating a protoplasmic artificial life form that was accidentally destroyed by Superboy in the accident that cost Lex his hair.
My Real Daddy: Sterling Gates for post-Crisis Kara. He gave her a personality that was palatable to the fans. Artist Jamal Igle moved her away from Ms. Fanservice, drawing her more like human being than a stick, and lengthening her skirt and putting shorts under it. Mark Waid may also count with his work with the character.
Replacement Scrappy: The Peter David version of Supergirl had detractors due to the angel angle, which was rather outside the norm for Supergirl (or even Superman). There are plenty though who greatly appreciated the series and wish the character wasn't Put on a Bus. This makes her above successors this trope as well.
Tear Jerker: The final arc of Linda Danvers' series, Many Happy Returns.
To explain: Linda, after years of trying to redeem herself for her past sins, is forced to send an innocent girl to her death just to save the universe from destroying itself (she had to send the Pre-Crisis Kara Zor-El back to her world so she could live her life and die fighting the Anti-Monitor), loses the eight years she spent in the Pre-Crisis Earth-One where she was Happily Married and had a child, and to top it off, she refused to send Kara back home unless her daughter Ariella was allowed to live. Only, Ariella was sent to live in the 853rd Century, as seen in DC One Million, and Linda had absolutely no idea where her daughter was.
Wangst: There was a brief attempt to turn the Post-Crisis version of Kara into a dark, brooding, whiny teenager. The fans didn't like it, and now she's upbeat again.
To say nothing of the Pre-Crisis version who had to spend years hidden in an orphanage and sabotage well meaning attempts by would be foster parents to take her in so she could keep her role as Superman's 'secret weapon'. Not to mention the first time she did get adopted it turned out to be by crooks who only wanted to squeeze money out of having her.
WTH, Costuming Department?: Her New 52 outfit gets some scorn for the odd design choices, notable the cut-out knees on the high boots, the strange red patch on the crotch that almost evokes the image of panties but just looks off, and the general armoured look that, like Superman's costume, doesn't really make much sense.
Idiot Plot: Ok, Kara. So you're personally responsible for losing the Omegahedron and decide to man up and go after it yourself. You heard less then 5 feet from you that without the Omegahedron Argo City has only a few days to live. So you spend the first day learning to fly, go to sleep in a park, then spend the second day making up your alter ego and going to school. Even when you get a solid lead on the Omegahedron and get up to go after it you sit down and complete the day, play soccer and shower because...the teacher and Lucy Lane said so? In addition, Selena also isn't the same one in the movie as she wastes her time using the Omegahedron to pick up guys instead of taking over the world.
Took the Bad Film Seriously: As incoherent as the film ended up being Helen Slater really worked on her role, earning herself a Saturn Award nomination. Even the reviewers who hated the film tended not to criticise her performance.