Schoolgirl Rival

In many anime, when a female protagonist moves to a new school, some girl, usually stuck-up, proclaims that the heroine is worthy of being her rival. The stuck-up girl will constantly engage the heroine, who usually doesn't pay attention to these efforts, or even the challenger.

Anime and Manga

 * Peach Girl: Momo and Sai for a bit.
 * A Little Snow Fairy Sugar had Greta constantly challenging Saga, who was usually distracted by the fairies' antics.
 * In ROD the TV, Anita was also challenged by one of her classmates.
 * Azumanga Daioh shows Kagura doing this to Sakaki upon the former's transfer into Yukari's class.
 * In a manner more in tune with the show's absurd nature, Tomo challenges everyone in whatever their best field is.
 * Natsume in Hidamari Sketch is Sae's self-proclaimed rival, but she's really doing it because she has a Tsundere crush on Sae.
 * Naturally occurs in Kujibiki Unbalance with the one-sided rivalry between Tokino and Renko- Renko holds a grudge against Tokino from elementary school, but Tokino is so ditzy she doesn't even realize Renko is antagonizing her.
 * Occurs in the backstory of Mahou Sensei Negima. When arrived as Mahora,   declared that they were rivals. As   was an Emotionless Girl at arrival, it took her several years to notice. This, of course, means that, given her lack of authority figures,   was basically raised by The Rival, to the extent that what emotions she did develop were developed in response to an annoying Ojou bothering her all the time. Explains a lot, really.
 * Kaede and Mana also seem to have this type of relationship; pretty much every time we see them interact they're either fighting or trying to one-up each other.
 * Although this has less to do with them being schoolgirls and more to do with the fact that they're both fantastically accomplished warriors.
 * With wildly contrasting styles, no less--one is a Ninja, the other a Gunslinger.
 * Eiko Magami and Biko Daitokuji from Project A-ko.
 * Chitose in Hitohira flips back and forth between being this to Mugi and being supportive.
 * In the supplementary manga of Magical Girl Lyrical Nanoha A's, no sooner did Fate become a New Transfer Student did she gain a schoolgirl rival in Alisa after the latter overachiever of a student learned that the former was way too advanced for the curriculum.
 * Seto no Hanayome has Lunar transferring into the school for the sole purpose of being Sun's rival. It actually starts a war among the students.
 * Erina Ogura in Kirarin Revolution is Kirari's "rival" because she's jealous of Kirari's increasing popularity - even though Kirari thinks them to be good friends.
 * In B Gata H Kei Kyouka instigates one with Yamada after Yamada beat her in a beauty pageant.

Video Games

 * Sakura Kasugano and Karin Kanzuki in Street Fighter Alpha. The storyline connecting them, and Karin herself, are Canon Immigrants from the Sakura Ganbaru! manga.

Western Animation

 * Kim Possible and Bonnie Rockwaller have this sort of relationship.
 * Johnny Test has Sissy Blakely, a rare example of the schoolgirl being rivals with a boy.