There could be a situation where Snape holds Harry Potter due to a misunderstanding. Snape thinks Harry has stolen a powerful and dangerous magical artifact from his private stores. He holds Harry to interrogate him. But as the story unfolds, they both realize it was a setup by another character who wants to drive a wedge between them and the wizarding community at large.
One scenario could be that Snape holds Harry Potter captive to protect him from a greater threat that he's aware of but Harry isn't. Maybe he has learned of a new dark wizard's plot against Harry and by holding him, he can keep him in a magically protected area until the danger passes.
One interesting scenario could be when Harry uses a clever spell in a new and unexpected way during their exchanges, catching Snape off guard. For example, he might use a transfiguration spell to transform something in the classroom into a distraction while he makes a quick escape from Snape's scolding.
One interesting scenario could be Harry using a charm to make Snape's potions ingredients misbehave during class. For example, the shrivelfigs start bouncing around the cauldron. Another could be Harry transfiguring Snape's cloak into a bright pink color while he's walking down the corridor. Or perhaps Harry using a spell to make Snape's voice change into a high - pitched squeak when he's trying to give a lecture.
One common scenario is Snape helping Harry with his Potions work. Since Snape is the Potions master, he might offer some extra guidance to Harry, which could be crucial for Harry's progress in school. Another scenario could be Snape protecting Harry from the bullying of other Slytherins. He might not do it in an obvious way, but still intervenes subtly.
One interesting scenario could be that during detention, Snape makes Harry clean all the potion bottles in the dungeon while lecturing him about the importance of precision in potions. Harry, in turn, discovers an old, forgotten potion recipe among the bottles and Snape is grudgingly impressed.
Perhaps they are both transported to an alternate dimension where they have no prior grudges. They have to work together to find a way back home. In this new situation, they learn about each other's strengths and weaknesses without the baggage of their past in Hogwarts, which leads to them getting along.
One scenario could be Snape using his wand to communicate with the dead. Maybe he discovers a spell that allows him to use his wand as a conduit to speak to Lily Potter. This would add a whole new layer to his character, as his love for Lily was a major driving force in his life.
One possible scenario could be in a potions class. Snape might be unfairly criticizing Harry's work as usual, but this time Harry decides he's had enough. He stands up, looks Snape in the eye, and calmly but firmly defends his potion - making process, explaining how he followed the instructions correctly. This could shock Snape into silence for a moment.
One scenario could be in a fanfic where Harry Potter reveals himself as a powerful wizard to a group of muggles who are being threatened by dark magic. He steps in, uses his magic openly, and they are both amazed and terrified. Another could be in a school setting within the wizarding world where he reveals a new, secret power he has to his friends to help them in a dangerous situation.
In a more dramatic scenario, during a battle at Hogwarts. Snape might find Harry injured and touch him gently to assess his condition. This could be a very poignant moment as Snape, who has always had a difficult relationship with Harry, is now showing a side of concern. It could also lead to a new level of respect between them.