A possible scenario is that Harry is poisoned by a dark potion. Snape, being a Potions master, detects the symptoms early. He then spends days and nights trying to find an antidote and finally manages to save Harry. This long and intense process could make them realize their feelings for each other. Or, Harry could be kidnapped by a group of rogue wizards who want to use his power. Snape tracks them down and battles them to rescue Harry. After the rescue, the shared trauma and Snape's heroism could lead to a romantic relationship developing between them.
One scenario is when Harry is in danger during a duel or a battle. Snape might step in to shield Harry from a dark spell. Another could be when Harry is being bullied by other students or facing unfair treatment from the faculty. Snape, in these fanfictions, uses his position to protect Harry. For example, he might give Harry private lessons to improve his skills and keep him out of harm's way.
One scenario could be when Harry is facing a difficult exam. Snape, despite his usual gruffness, takes Harry aside and tells him to focus on what he's learned rather than stressing about the outcome. This simple yet firm advice calms Harry's pre - exam jitters.
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.
Often, Snape will single out Harry in front of the whole class. He might publicly criticize Harry's work or attitude. For example, when Harry is assigned a group project in Potions, Snape might give Harry extra tasks or set higher standards just for him. And during exams, Snape could be extra strict when grading Harry's papers.
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.
In some fanfictions, there could be a situation where Snape's true motives are being misinterpreted by the wizarding community. Harry, who has a deeper understanding of Snape's character, defends him. He could talk about Snape's unrequited love for his mother and how that love led Snape to make difficult choices. Harry may also point out that Snape was a key figure in the downfall of Voldemort, and without him, the outcome of the war would have been very different.
In such a fanfiction, perhaps Harry discovers some dark secret of Snape's that makes him decide to take action against him. Maybe Snape has been plotting something truly evil all along, and Harry finally uncovers it and has to destroy Snape to protect Hogwarts and his friends.
Perhaps in this fanfiction, Harry submits to Snape in order to gain access to some important information. Snape might be in possession of secrets about Harry's family or his own powers that he refuses to share unless Harry shows complete obedience. So, Harry decides to submit to Snape's demands in order to uncover these mysteries that are crucial for his journey as a wizard.
One common scenario is that Snape saves Harry's life in a more personal way than in the original story. This act of heroism makes Harry see Snape in a new light and starts the journey towards friendship. Another is when they discover a common interest, like a love for a particular type of magic or potion - making. This shared interest bridges the gap between them.