One of Miss Minchin's chief secret annoyances was that she did not speak French herself, and was desirous of concealing the irritating fact. (2.24)
This is the moment when Miss Minchin really starts to dislike Sara—because she has a desired skill (that pretty, fluent French-speaking!) that Miss Minchin doesn't have. Notice how the jealousy is way more about Miss Minchin's inadequacies than what Sara actually has?
Lavinia, in fact, was spiteful. She was inordinately jealous of Sara. Until the new pupil's arrival, she had felt herself the leader in the school. (4.6)
Your days are numbered, Lavinia. Next time, try being nice to people. Seems to work for Sara.
"She's ridiculous without being rich," Lavinia sniffed. (6.4)
Lavinia can't find enough bad things to say about Sara, even though she hardly knows her. Check out the fun way that Burnett makes Lavinia "sniff." Can't you just see her little nose stuck up in the air?
"I suppose she thinks she could be a princess if she was a beggar," said Lavinia. "Let us begin to call her Your Royal Highness." (6.13)
This whole "royal princess" thing kind of backfires on Lavinia when Sara takes it to mean that she should act like a princess in the generous and proper sense, not in the spoiled rotten sense.
"… And it's ridiculous that she should look so grand, and be made so much of, in her rags and tatters!" (16.12)
Even when Sara's poor, Lavinia feels threatened by her. Hmmm, looks like Sara isn't just a spoiled little princess after all.
"Perhaps the diamond mines have suddenly appeared again," said Lavinia, scathingly. "Don't please her by staring at her in that way, you silly thing." (16.83)
Oh, Lavinia. We would almost feel sorry for you, if you weren't such a waste of paper. (Just kidding. We need you around so we can see how much we like Sara.)
"You will never see your companions again," she began. "I will see that Ermengarde and Lottie are kept away--" (18.71)
Miss Minchin is truly spiteful, but thankfully, Sara's got a lawyer on her team. Always good to have a lawyer around!
"The diamond mines!" Miss Minchin gasped out. If this was true, nothing so horrible, she felt, had ever happened to her since she was born. (18.54)
Hah! Serves you right, you mean, nasty thing. Miss Minchin will never get her hands on that money now. And what's even better is she has to watch Sara's every move, since they're still living right next door.
"…The fact was, she was too clever for you, and you always disliked her for that reason." (18.82)
Miss Amelia rarely appears in the story, but at the end, it's clear that Miss Minchin (and the reader) has underestimated her. She knows what's up. Too bad she didn't speak up earlier, when it could have done Sara some good.
The next morning, Miss Minchin, in looking out of her window, saw the things she perhaps least enjoyed seeing. (18.24)
Miss Minchin will just have to deal with that envy and regret boiling in her chest forever. We only wish we were going to be around to see it.