Best Known for the Fanservice: The mizuage scene was so controversial that it's probably the best-known moment of the film (and thus what makes casual observers remember most about geisha, much to the chagrin of real geisha).
Broken Base: The casting of the three Chinese actresses (though Michelle Yeoh is Malaysian with Chinese descent) as Japanese characters sparked furor among the Chinese and Japanese audiences. While the Japanese are upset that no Japanese actress got any of the main roles, the Chinese were very hostile about it which doesn't help that the fragile historical relationship between the two countries since World War II which the movie is set in only fanned more flames. However, several people such as Ken Watanabe defended the cast, citing the actresses' performances was the reason for the casting.
Draco in Leather Pants: Aside from being enemies of Sayuri, Hatsumomo and Pumpkin earned this kind of attention from the movie's audience.
Pumpkin. Not as pretty as Sayuri, not as talented as Sayuri, not as smart as Sayuri, not as lucky as Sayuri, and browbeaten by her supposed mentor Hatsumomo into the bargain. Oh, and Sayuri is unwittingly responsible for Pumpkin's downfall.
Hatsumomo as well. While she is a sadistic bitch, like Sayuri she was not free to be with the man she loved- and was even harshly slut-shamed for it.
Rooting for the Empire: Hatsumomo and especially Pumpkin get this treatment from some viewers. Nobu gets a similar reception, if only because Sayuri reacts to his desire to be with her the way most heroines respond to the unwanted advances of a villain, when she was the one who started their relationship in the first place. For Pumpkin and Nobu, also see Unintentionally Sympathetic below.
Chiyo was sold to an okiya, a very harsh, bitter place where she was abused relentlessly; lost both her parents; and was separated from her sister, whom she never saw again. And that's only the beginning!
Mameha furiously lambasting Sayuri, who was forcibly stripped by the Baron because he wanted to see her naked before her Mizuage, telling her that everything will be ruined if she's found to be "worthless" and storming out of the room. Sayuri can only mutter a, "I'm not worthless" whilst barely holding back tears.
Pumpkin's revenge against Sayuri is meant to be a major Kick the Dog, but as she points out to Sayuri, Sayuri took Pumpkin's life goal of becoming officially adopted by the okiya and becoming one of the most successful geisha in Gion, and she didn't really even want that; it was all just "a stepping stone" to get to the Chairman. Under those circumstances, it's easy to feel like Pumpkin's revenge was justified, even if Sayuri hadn't meant to ruin Pumpkin's life.
Similarly to the Pumpkin situation, Nobu's reaction to Sayuri could be seen as extreme, but she did make him believe she loved him, under instruction from Mameha, so his feelings of betrayal when she and the Chairman get together is pretty justified, especially as the Chairman pretty much orchestrated the whole thing. However, Nobu does also come off as Entitled to Have You, doing favours for Sayuri in order to make sure she's so indebted to him that she has no choice but to accept him as a danna, and he repeatedly lambastes her in the book for already having a danna in General Tottori, which Sayuri had no choice in.
Unintentionally Unsympathetic: Sayuri about three-quarters through the book. Though she starts off very sympathetic in her Trauma Conga Line and Break the Cutie moments, it becomes very clear that she's willing to do pretty much anything just to get with the Chairman. This includes screwing over her closest companions and showing no remorse for doing so. Values Dissonance also ensues once it's revealed that the Chairman was married the whole time.
The Woobie: Chiyo, at least until Mameha decides to take her on.