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.
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.
One scenario could be that Voldemort has a change of heart. He might have had a vision or a revelation that made him realize that Harry is more valuable alive and not under the cruel treatment of Vernon. Another scenario is that Voldemort wants to manipulate Harry. By rescuing him from Vernon, he can start to build a relationship with Harry based on gratitude. This way, he can later persuade Harry to join his side or use Harry's power for his own purposes.
One common scenario is after a battle. Harry is injured, and Snape, despite their differences, steps in to heal him. Maybe it's a fight against the Death Eaters where Harry is seriously wounded, and Snape realizes he has to put aside their enmity for the greater good.