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 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.
Often, in the fanfic, it could be that Harry is injured during a Quidditch match. Snape, despite his dislike for Harry in some ways, realizes the importance of Harry to the wizarding world and heals him. There could also be a situation where Harry is affected by some dark magic in the Forbidden Forest. Snape, who has a deep understanding of dark magic, is the only one who can counter it and heal Harry. It might also happen that Harry is suffering from the after - effects of using powerful magic, and Snape uses his own unique magic to heal Harry's drained energy.
One common scenario could be that Harry is caught sneaking around the Slytherin common room or stealing potions ingredients from Snape's stores. So, Snape and Draco might make him do extra detention tasks like scrubbing the cauldrons for hours without magic. Another situation could be if Harry is suspected of cheating in a Potions exam. Draco might report it to Snape, and they could punish Harry by making him redo all his Potions experiments from the semester.
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.
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 common scenario is Snape protecting Harry from the Dark Arts attacks within Hogwarts. Another could be Snape helping Harry with his studies privately, especially in potions where Snape is an expert.
One common scenario is when Harry has a nightmare about his parents' death. He might then seek out Snape, knowing that Snape was also connected to his mother. Snape could offer some memories or stories about Lily that would bring Harry a sense of comfort. Another scenario could be when Harry is struggling with his identity as the Chosen One. Snape, who has his own complex identity within the wizarding world, could provide some perspective and comfort to Harry.
Well, in some fanfictions, Snape's discipline makes Harry more determined. Harry, facing Snape's strictness, learns to be more cautious in his actions and words. It forces him to think twice before doing something, which in turn helps him grow into a more responsible wizard.
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 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.