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.
Alternative Character Interpretation: Most fans of this show clearly identify with Sara and see her as a strong character. However, according to an interview with Fumio Kurokawa, the director, she wasn't really meant to be a role model or a Mary Sue (in fact, he views her as rather passive), and both Miss Minchin and Lavinia have admirable qualities and are not meant to be one-dimensional villains. This is partly supported by the little that we see of the two characters' backstory - Miss Minchin's difficult childhood, and Lavinia's parents.
Fridge Logic: Sad to say, but even a series of this quality is not immune to the Curse of the Refrigerator. Near the end of the series, the Deus ex Machina Mr. Carrisford, feeling sorry for the girl in the attic next door - whose identity he does not yet know - asks a dressmaker to make her some nice clothes. This dressmaker is the same one who sold Sara and her father her precious doll, Emily, right at the beginning, and he has already met her after her fall from grace (episode 34) and knows that her father is dead. A simple question from Carrisford at this question would have saved him - and Sara - several days of misery. Of course, this allows The Reveal to be postponed for dramatic effect, and sets up the events leading up to Sara's expulsion, which are among the most moving in the series.
Fridge Brilliance: In the live-action. Kaori is an original character to the original story and her emotionlessness and snarkiness certainly was out of place compared to the other characters. But when you think about it, she is the few people who openly says they doesn't like the way Maria treats Seira and can get away with it and she questions Seira why she stays even when she's treated terribly by Director Chieko and Maria ... which are often the same thoughts of the viewers themselves. You could say Kaori is the Author Avatar for the audience.
Germans Love David Hasselhoff: Saudis and Filipino fans love Princess Sara that re-runs of them are not uncommon. The Filipinos even made a live-action movie back in 1997 based on the anime!
Same for France. It's one of the numerous anime originally aired in The Eighties that gets the most re-runs.
In the live-action, wearing a Meido maid outfit only increases Seira's moeness tenfold.
Nightmare Fuel: In episode 16, "Lottie's Adventure", there is a scene in which Lottie has a nightmare and runs to Sara for comfort. This nightmare involves deformed trees that turn into Miss Minchin and Lavinia's faces, and a wolf that tries to eat her (Lottie plays the role of Little Red Riding Hood in the dream.) Pretty scary stuff for children, and it was censored in the French dub.
Purity Sue: In the original novel, though Sara is kind, generous, imaginative and clever, she is not portrayed as particularly pretty, can be critical and judgmental of others, and has a short temper. In the anime, though, she is almost saint-like, frequently described as beautiful (the lyrics to the French and Italian dubs state this explicitly), and the only times she loses her temper are when she defends Lottie from Lavinia's bullying. That doesn't stop her from being an immensely likeable character, though.
Toy Ship: It's made pretty obvious to the viewer that Peter (clearly a pre-teen) has a schoolboy crush on Sara (aged 9), and he often helps her out of tight spots, including aiding her in sending a letter to the Bombay police to enquire about her father's death; the scenes of the pair talking together on the bridge exemplify this trope. Towards the end, he actually takes her into his home when she is cast out of the Seminary.
The Woobie: Sara deserves the most hugs out of everyone in the show. Kaito from the live-action drama needs some too.