Azmaria in Chrono Crusade, the Token Mini-Moe with spiritual powers that allow her to heal others and fly...and not much else. In the anime she's mostly there to look cute and get in trouble, although she does serve as a peacemaker of the group. In the manga she behaves this way at first, but through Character Development she slowly becomes stronger and more determined—enough that, during the final battle, her ability to stay calm under pressure holds the rest of the team together during a crisis and her spiritual knowledge serves a key part in stopping the Big Bad and his plans.
"G-3" Jun the Swan from Science Ninja Team Gatchaman. A Trope Codifier for Anime/Manga characters. Primarily, the Token Girl from being on an all male team. She can be seen as either an Action Girl who is self-sufficient from owning and running her own snack bar and a skilled fighter that can knock thugs unconscious with the same ease of her male companions, or a victim of Chickification and occasional Distressed Damsel when the story calls for it to give Ken/Joe some hero spotlight moments. Also, since she's the technical and demolitions expert of the team, she is arguably better at being The Smart Guy than the show's actual smart guy.
Sakura of Tsubasa Reservoir Chronicle is both a straight example and a subversion, since that character had already appeared in another work by the same authors, where she was The Hero. She kept exactly the same personality, but with a serious downgrade in magic abilities and three fighting experts as companions she transformed into a Chick. She also regularly lampshades the fact that she feels useless in the group because of her inability to fight. Later she upgrades to being an Action Girl once she gets her hands on some guns and A Day in the Limelight. She also doubles as The Empath, the Oracular Urchin and The Messiah. Later, in Infinity country, once a certain member of the group has left, she arguably becomes The Hero and (literally) The Chess Master.
Botan, Keiko, Yukina, Shizuru and Atsuko from YuYu Hakusho. All five have needed to be saved at some point (manga-only, in Atsuko's case), but are generally side characters just outside of the Competence Zone. They're all tougher than the normal Chick (Botan, Yukina and Shizuru have more-or-less useful powers too) and can hold their own against humans, but they still can't reach Badass Normal level when facing demons. Or can they?
Judy in Night of the Living Dead is rather noteworthy in that her character was created specifically because the crew were enchanted by how cute the actress was.
In Star Wars, both Princess Leia and her mother Padme Amidala are the Action Girl version of the Chick.
In Attack of the Clones, Padme frees herself from capture by the Geonosians and takes part in defending the Jedi against the droid army attack and the pursuit of Count Dooku.
Leia is in firefights in all of the original trilogy's films, and tries to rescue Han Solo from Jabba the Hutt, later killing Jabba by herself.
There are a variety of psychology/morality puzzles that involve four people who get themselves into a situation where a series of events occur, usually culminating in tragedy, and the listener is charged with putting the characters in order from who acted the most "right" and who acted the most "wrong". Of the four, only one will ever be female, she will always be The Chick, and usually she's the one who suffers the tragedy. Where the listener places her in the morality spectrum represents their views on women, regardless of what she does in the puzzle story; the other three usually represent specific virtues or behaviors, but the female represents only women as a whole. With most of these puzzles, putting the female character at anything lower than the Most Right (or, less often, Second Most Right) will label you a misogynist.
While Animorphs fans have had some trouble agreeing on who's actually The Chick, Rachel has never been a contender. She's beautiful, her hobby is gymnastics, and she loves both shopping and her boyfriend, but she's The Big Guy, an Action Girl who becomes a Blood Knight.
Jacyl from Black Dogs is extremely fond of clothes and beauty products, and one character remarked that she was found while her house was burning down, and that she was trying to rescue her wardrobe. Naturally she's one of the more levelheaded and pacifistic characters in the book.
Journey to the West features a very... odd subversion of this trope - the monk Xuangzang is so perfect an example of The Chick he's (note the masculine) sometimes even portrayed as such. This, of course, leads to tons of Ho Yay with the other main character of the story, SunWukong. Given his overall uselessness, stupidity, and propensity to be captured by anything under the sun, it's no wonder why most everyone who reads the story kinda really hates him. In a double subversion, it's also as become common to protray Wukong as the girl, just to make him (her?) seem more awesome.
Jenna Heap from Septimus Heap. She is usually the one who cares about people and prevents the others from ignoring them in their actions.
Live Action TV
Detective Irene Daniels of The Closer, who isn't seen in too much physical combat (although she can certainly hold her own), and is much more girly than the rest of the all-male Priority Homicide Squad (besides, of course, Deputy Chief Brenda Leigh Johnson). However, she is apparently brilliant at forensic accounting - tracking down relevant documents relating to murder victims and suspects.
Gwen from Torchwood prior to becoming second-in-command to Captain Jack. Even before then she could hold her own in combat.
Robin Quivers of The Howard Stern Show.. Her official position is to deliver the news, yet often she's the heart of the show.
Rika of Phantasy Star IV is The Chick for most of the game once all of the permanent, non-optional members are in— she's even briefly The Chick in a Power Trio. However, she's much more well-balanced than the typical RPG example: all of her magic— all of it— is healing or status magic, but she's a powerful melee fighter and her physical attacks are second only in damage output to Chaz once he gets the Elsydeon. Before that, they're about equal.
Princess Peach from the Super Mario Bros. games. Even in her solo game, Super Princess Peach, her abilities are very stereotypically feminine. More evident in Super Smash Bros. Brawl - in The Subspace Emissary she makes herself useful by breaking up fights and turning potential enemies into friends, mysteriously producing tea, in contrast with the Action Girls Samus and Sheik.
Sailor Moon herself is pretty firmly The Chick, despite being the main character andThe Messiah. She acts as the team's moral and emotional pillar, and is least often the one to take frontal assault duties when fighting various enemies, generally doing her best to simply survive until she has a chance to whip out a Finishing Move. This also makes sense when you consider that Sailor Venus was originally the leader in some incarnations.
Ami (Sailor Mercury) is sweet, shy, bookish and has defensive abilities based on water and ice, so you'd expect her to be a secondary Chick after Moon. Instead, she's The Smart Guy, since she also happens to be a Teen Genius.
In Mass Effect, both Tali and Liara serve as both The Chick and The Smart Guy. In the sequel, Tali and Kasumi both serve as Smart Guys, Miranda serves as The Lancer and a bit of The Smart Guy and Jack serves as one of The Big Guys.
Danny Phantom has Jazz. She doesn't join the team at first, but she remains a kind and compassionate person who sees the good in others and drives them to better themselves throughout the series, even if others can be a tad annoyed with her because of it.