Take the two Harry Potter characters you think least likely to have a fling on the floor of a classroom. Throw them together in detention and make it happen. Whether you decide to somehow make it plausible, or silly, or outrageous, is up to you. One of them can be giving detention to the other, both can be in detention, or one can be in detention and the other mocking them - it's your call.
The two least likely to have a fling? So many options! I decided on Flitwick/Mrs. Norris... Kind of.
Title: An Evening Fling
Pairings (if any): Flitwick/Mrs. Norris. Kinda.
Author's Notes: I counted in my 30 minutes Googling to find a Latin word to use as a spell.
Flitwick sat at his desk, grading scrolls from today's class. He shifted on the pillows stacked on his chair so that he could glance at the clock. Just after eight. This would be detention time if he had had given any out in the past week, but luckily his classes had all gone surprisingly smoothly, and so the small professor hadn't had to give out any punishments.
Finishing with one scroll, he placed it on the stack and reached for another. His moment of wandering attention allowed him to catch sight of his visitor standing in the doorway.
"Hello, Mrs. Norris!" he nodded to the scruffy feline. While others, mainly students, were wary of the cat (and her owner), Flitwick rather liked the small animal. "Come for a visit, have you?"
The dust colored cat meowed at him as she stepped into the room. In a rather meandering path, she eventually made her way to his chair. A few more meows were all it took to get the small teacher to jump down off his pillows and then off the chair. He crouched next to her and give her back a fond pat. "You need to head off, Mrs. Norris," he told her kindly. "I can't concentrate with you making all this racket."
The cat meowed rather insistently at him.
"Well..." he drew the word out, glancing first towards the door then at his desk (not that he could see the waiting scrolls on it from this angle). "Okay, but only once tonight. O.W.L.s and N.E.W.T.s are coming up, and I have a lot of grading to do." Nevermind that she couldn't understand him. "Okay, get ready!" The professor reached into his pocket as she rubbed against his leg.
"Fling the cat!" he cried happily, then waved his wand. "Iacio Mrs. Norris!"
The skinny cat gave a happy little yowl as the professor's magic lifted her into the air and swiftly sent her in a controlled arc, keeping her safe and landing her on her paws outside the door. It was even better, even further, than she could imagine leaping! Her tail twitched in happiness and she meowed at Flitwick one last time, then returned to her rounds. With a chuckle, the little professor climbed back up onto his seat and went back to work.
It's very hard to resist doing little edits after the 30 minutes are up.