I've just gone through your anti snape tag and I was wondering why, in all your many valid points that I completely agree with, you hadn't mentioned the incident where child! snape dropped a heavy branch on petunia? [or am I remembering this wrong] x
I think I never mentioned it is because I focused on his Hogwarts years and afterwards (when he was a death eater and when he was teaching). but this has gotten me to think a bit (why it took me a while to answer this) and I think the thing with Snape and Petunia really shows how he did find Muggles to be beneath him even before Hogwarts.
when he first meets Lily and Petunia he says this when Petunia accues him of spying (which he kinda was doing)
“‘Haven’t been spying,’ said Snape, hot and uncomfotable and dirty-haired in the bright sunlight. ‘Wouldn’t spy on you anyway,’ he added spitefully, ’your’e a Muggle.’
Though Petunia evidently did not understand the word, she could hardly mistake the tone.” ~Harry Potter and the Deathly Hallows
Right here, Snape’s tone and everything just shows how much he hates Muggles and by extension Petunia. Yes, it wasn’t very nice on Petunia’s part to accuse Snape of spying despite it being an accurate accusation. But at the same time, Snape just called his sister a witch and claimed to be a wizard which would seem really far-fetched and could be seen as a slight against Lily whom she did not hate this point. And I think Petunia is the kind of person who would judged someone based on their appearance and Snape’s appearance wasn’t very good (which isn’t very right either). At the same time there was really no reason for him to call Petunia a Muggle in a very spite way, making it obvious that he found it to be a bad thing to be a Muggle and saw it as an insult.
Now to what you were talking about in your ask.
"Then a small rustling noise behind Harry made him turn: Petunia, hiding behind a tree, had lost her footing.
'Tuney!' said Lily, surprise and welcome in her voice, but Snape had jumped to his feet.
'Who's spying now?' he shouted. 'What d'you want?'
Petunia was breathless, alarmed at being caught. Harry could see her struggling for something hurtful to say.
'What is that you're wearing, anyway?' she said, pointing at Snape's chest. 'Your mum's blouse?'
There was a crack: A branch over Petunia’s head had fallen. Lily screamed: The branch caught Petunia on the shoulder, and she staggered backward and burst into tears.
'Tuney!'
But Petunia was running away. Lily rounded on Snape.
'Did you make that happen?'
'No.' He looked both defiant and scared.
'You did!' She was backing away from him. 'You did! You hurt her!’
'No—no I didn't!
But the lie did not convince Lily: After one last burning look, she ran from the little thicket, off after her sister, and Snape looked miserable and confused…” ~Deathly Hallows, bolded mine.
This passage says a lot about Lily, Petunia, and Snape. It is still obvious here that Lily still cared and loved her sister and wanted her to be included. It, also, shows the hatred between Snape and Petunia how both don’t like each other. Here, Snape is the first one to insult Petunia (though it’s wrong of her to spy like it was with Snape, and I believe Lily would have loved to include her based on her reaction to seeing her but Snape wouldn’t want her based on his reaction). then Petunia throw an insult back at him leading to him causing the make a branch fall on her. Yes, words hurt but it is never right to lash out and hurt someone physically. He could have seriously injured Petunia, even killed her if the branch fall at the right angle. And from his reaction afterwards, it is obvious that he didn’t feel remorse and felt he was right (at least that’s how I see it). He directly lied to Lily about hurting her sister and it scared Lily (with good reason).
The last thing I want to say is from the scene on the Hogwarts express right before James and Sirius said anything to them.
"Snape slid open the compartment door and sat down opposite Lily. She glanced at him and then looked out the window. She had been crying.
'I don't want to talk to you,' she said in a constricted voice.
'Why not?'
'Tuney h-hates me. Because we saw that letter from Dumbledore.'
'So what?'
She threw him a look of deep dislike.
'So she's my sister!'
'She's only a—' He caught himself quickly; Lily, too busy trying to wipe her eyes without being noticed, did not hear him.
This is really important. Because it shows how he doesn’t care that Lily is upset that Petunia hates her; he just doesn’t seem to understand why Lily cares about a Muggle (because he was going to say she’s just a Muggle). But Lily does care because Petunia is her sister and they were close once and part of the reason they fall apart is because of Snape (his obvious dislike of her, insulting her because she is a muggle, causing a branch to fall on her then denying it, and going through her private things—her letter to Dumbledore). And right here shows that he doesn’t care that Lily’ relationship with her sister is falling apart because Petunia is just a Muggle and not important. And the fact that he had to catch himself from saying that, means he knows that Lily doesn’t like the way of thing and he has to hide it from her but he still has that prejudice against Muggles. And he is actually very like that Lily did not catch what he was going to say because I pretty sure she would have been really mad about it and could have made their friendship end early then it did in canon (at least I think so).
I think overall, it does show that Snape was prejudice against Muggles from the beginning even before Hogwarts and the main reason he didn’t like Petunia was because she was a Muggle.
and this is proabaly more that you wanted but your question really got me thinking about this and I think does give a good insight to Snape’s character and how he was always prejudice just that he made a exception for Lily.