Snow White yearns for more than Prince Charming to rescue her. She wants love, understandable given that she lost her parents. Her household chores when coming upon the dwarves' cottage are voluntary: she empathizes with what she thought were parent-less orphans.
Cinderella is trying as much as she can to buck the system in an abusive environment: one that isolates her from others and in an era that doesn't offer much opportunities for women other than housekeeper and governess. Her sarcastic rant at a bell ringing, the fact that she frees mice from traps and dresses them, and her arrival at the ball are acts of rebellion.
Aurora is a victim of social conditioning: naive and isolated by understandably over-protective guardians that haven't allowed to develop outside of the idealistic persona they've held her up to.
Ariel is an outcast yearning to go somewhere where others will understand her, she's plenty interesting, and willing to stand up for herself (she did kill the eels of Ursula).
Belle doesn't suffer Stockholm Syndrome and is a proud non-conformist who is brave enough to stand up to the popular town misogynistic hunk and a beastly Prince. Her actions show Gaston (at least to the audience), just how bad he is and encourages change in the Beast.
Jasmine being allowed to pick a suitor to be married to, was progressive for their time and place. She is intelligent, courageous, witty, and kind; she doesn't easily give in to Aladdin's/Prince Ali's courting of her so easily. She is also a terrific actress.
And that is just the ones whose characterization is relatively simple and has more straightforward writings. Is Mulan's motivation for disguising as a man and ran to the army is for her father or for herself? Some mix of all of the above? (this was also brought up in-universe by Mulan herself) Is Merida a normal teenager who made a mistake or a horribly spoiled brat? Is Elsa's power an allegory for mental disorder, her sexuality, something else or all of the above?
Ariel. Fans are a bit divided on whether she's "an idiot for throwing her life away to be with someone she doesn't know" and should be regarded as poorly as the previous three Disney Princesses (who haven't aged well), or she should be commended for being "the first Disney female lead to have an actual personality and be the one to save her prince first."
In a similar pattern to Ariel, Merida is criticized for her impulsiveness and general teenage attitudes. Some find her to be refreshingly normal and realistic for her age as well as a certified badass, others find her to be just an insufferable brat with a remarkable callous streak.
Pocahontas. Fans either think a decent role model that stand out due to the Green Aesop of her movie despite the glaring historical inaccuracy or an incredibly shallow character with little personality, preachy message and an insult to the real Pocahontas. And then there are people who think that while a passable character, she and her movie is a total letdown compare to the previous installment's incredibly flawed protagonist Simba from The Lion King.
Over the inclusion of characters who do not seem to fit with the line as easily as characters like Aurora or Rapunzel; that is, Pocahontas, Mulan, and Merida for being too athletic, not girly enough, and/or not a princess.
Similarly, if Elsa joins the line-up, a lot of people feel that it is insulting somehow to being her queenship.
Moana fans have been somewhat divided over the prospect she could join, though most seem to support it. The main debate is if she's technically a princess or not. She's a chief's daughter, and her culture lacks the connotation of princess. On the other, even Maui points out that she fulfills the qualifications, brave, strong, sings, has an animal sidekick, and her title is the closest to a princess anyway. A concern for fans is her body type, which was made to go against the mold as she has a lot of muscle and built to look like it.
Alternatively the exclusion of certain characters is this. For example, Esmeralda from The Hunchback of Notre Dame (who actually was a Princess character early in the franchise) or even Nala from The Lion King are fan-favorites but are not in for different reasons, however people often point at non-royalty like Mulan and question this.
Disney fans in general often feel this way about the princesses, particularly fans of the heroines. The Disney Princesses receive more merchandise and materials for children *and* adults than any other characters. Some perceive this favoritism to be adversely affecting their own favorite characters.
Disney Palace Pets is Snark Bait amongst much of the older fanbase. The designs and premise itself is controversial, though people do tend to enjoy the various unused concept designs.
On a more interesting point, there's the idea of whether Sofia and Elena should be included in the lineup. This stems from how the latter is a spinoff of the former, which is connected to the franchise due to the cameos the princesses made in various episodes. Both Sofia and Elena fit the mold of brave, altruistic princesses who are independent and caring, and are generally well liked. However, there's also the issue of them being exclusive only to tv, and never starting out in theaters. One side thinks that a princess has to have a theatrical movie to be considered part of the lineup, especially since Sofia is much younger. The other side thinks that their cultural heritages and diverse backgrounds should qualify them for the lineup regardless of theatrical status.
Critical Dissonance: Despite its critics, the franchise is one of Disney's biggest and most profitable.
Ensemble Dark Horse: A 2016 poll ranked the popularity of the Disney heroines. Out of the princesses, Cinderella and Aurora - the two most often attacked for being 'anti-feminist' - placed first and third respectively (Elsa from Frozen placed second).
Fanon: Honey Lemon fans believe she could be part of the lineup, even saying she's probably related to Rapunzel to confirm her royal blood.
Minority Show Ghetto: It was noted that the darker princesses' merchandise didn't seem to sell as much as the white princesses'. Whether this was due to the popularity of the movies themselves, or that less merchandise over all was produced is up for debate.
Moe: Cuteness is often heavily associated with the princesses, as many of them are The Ingenue or have qualities like that. Snow White and Rapunzel seem to be the ones who provoke this reaction the most.
Most Wonderful Sound: Most of the princesses have beautiful singing voices and have their vocals provided by top drawer talent. As a result, their songs in their movies are usually quite pleasant to listen to.
Older Than the Demographic: The characters are all teenagers and the original films are aimed at general audiences, however the Disney Princess line is firmly aimed at little girls ten and under.
Periphery Demographic: While the franchise started out targeting young girls exclusively, Disney eventually realized the untapped market of *older* fans who had grown up with the princesses and wanted merchandise for themselves who had always been there to buy some of the more reserved dolls or prints. Now Disney has whole lines catering to them, to expensive fashion dolls, women's clothing, and more.
Tastes Like Diabetes: The extra merchandise and sequels are loaded with saccharine morals, songs and messages that tends to be unpalatable to anyone but very young girls.
Testosterone Brigade: The especially Ms. Fanservice characters—Ariel, Jasmine, and ostensibly Pocahontas—invite a lot of male fans, but many little boys grew up with crushes on one of the princesses that stuck.
Many of the girls coloring and style is changed for the merchandise, such as Aurora's lighter hair, Cinderella's dress changing from silver to blue, her hair changing from strawberry-blonde to bright blonde and so on.
After Disney unveiled new hairstyles and dresses, some people weren't very happy about that.
Aurora and Tiana graced the wrappers of some dipping candy. Aurora got the packet of vanilla-flavored sticks. Tiana got the packet of watermelon-flavored powder. This was met with some amount of controversy.
The new designs attempted to homogenize the general look of the Disney Princess line and the different art-styles, which led to everyone either getting paler or darker, the brightness being turned up to max, and some of the efforts to make the art seem more detailed really failed to the point where dark-eyed characters seemed blue-eyed. The homogenization of art style, however, led to the faces looking more Caucasian than anything. This sparked a pretty big backlash even outside of regular fandom circles.
Values Dissonance: One justification of the Real Women Don't Wear Dresses criticism. Not only were the original fairy tales written centuries ago, but the time periods in which the earlier Disney Princesses were made had the "demure-but-hard-working" type as the epitome of womanhood in mainstream America. Naturally, this flies up the head of many modern fans who just peg them as "weak" and "whiny".
All of the Disney Princesses are treated as such. The older princesses receive the most disdain, as the time period their movies were made from put them in the passive role that would not be acceptable in a female role-model today. Even the modern princesses are often closely scrutinized and found unworthy, particularly Ariel. While there is some truth to the criticisms, the biggest criticisms lobbied at the girls tend to twist the actual events of the movie. For instance, the common criticism that Ariel gives up her life at home for a man isn't exactly true—Ariel clearly desired to live with humans long before she knew who Eric was (her famous musical number was even all about it!), he just was the catalyst to actively going out, as well as blaming Snow White and Aurora for "not saving themselves", despite the fact that both were under spells against their will and could do nothing about it.
Disney draws a lot of ire from the older fans when Mulan is put in a feminine dress (particular the pink one with make-up that she felt uncomfortable in). However, as with the above case, even those criticisms are often twisted—Mulan is also clearly uncomfortable wearing armor and hiding herself that way, and did like looking cute in the pink dress even if she disliked the prospect of an Arranged Marriage. The happy medium is usually the far more practical (and plain) dress she wears at the end of the film.