It spread faster than wildfire, as things of this nature often do at Hogwarts.
Neville Longbottom's greatest fear was Professor Snape.
He heard them snickering in the hallways as he passed by, some because the descriptions of his grandmother's clothes on the potion's professor, others by the fact that he was afraid of Snape, and really, he was a nasty, greasy old bat, but to be your greatest fear? How ridiculous was that?
But none of them understood. None of them knew.
Snape constantly criticized and belittled Neville, always criticizing every little thing he did wrong and saying it loud enough to the whole class to hear. And that was bad, yeah, but that's not why he was Neville's greatest fear. Everyone expected him to be great, just like his parents, but Neville let them down again and again as he failed at nearly everything he did. His Gran was the worst, expecting him to be just like her Frank. Neville tried, he really did, but he could never live up to her expectations.
To Neville, Snape symbolized all of his flaws, everything he couldn't be and couldn't do, no matter how hard he tried. Snape made him feel useless, like he'd never do well at anything and that everyone's expectations were all piled onto his shoulders at once, and they'd come crashing down and bury him at any second.
And Neville hated feeling like that, more than anything in the world.