"Slytherin vs. Gryffindor"
The squabbling between Slytherin and Gryffindor began with Salazar and Godric, and no professor at Hogwarts since has been either arrogant or foolhardy enough to believe that it will stop during the next term. It is tolerated, like the shifting staircases and Peeves, as one of the inevitable nuisances of school days.
However, most of these professors have also known that, at times, darker forces are at work in the rivalry. They watch it closely -- always in frustration and sometimes in humor, but every so often with caution.
Half a century ago, some of them noted how many Gryffindors had made mistakes in Potions (burnt fingers, fainting spells), or had fallen from their brooms during Quidditch (two broken arms and a crushed ankle in the first match alone). They were astute enough to realize that the cause might be found in Slytherin; unfortunately, none of them could imagine that the wrongdoer might be penniless, brilliant Tom Riddle. If they'd investigated just a bit more, tragedies might have been averted.
Or perhaps not. Tom was always clever, and from childhood he remembered what many adults forget: School can be the training ground for war.
The bitter feud between Severus Snape and James Potter, Sirius Black, Peter Pettigrew and (to a lesser degree) Remus Lupin was the source of considerable irritation and worry before the Prank. (It was legendary among the staff, too, at least for a few years.) Afterward, the four Gryffindors were roundly punished and so abashed that they became far less rambunctious than they'd been before. Minerva McGonagall was heard to remark that James Potter had actually become a pleasure to teach, which made one wonder whether someone had at last answered her prayers and developed a spell to deal with troublesome boys. Most of the professors saw the incident as regrettable, but perhaps necessary, as the Gryffindor boys had learned their lesson. Only Dumbledore saw how much more secretive and angry Severus Snape became afterward, and how his anger was no longer focused strictly on his rivals but instead had acquired a far greater reach.
Only Dumbledore and one other, that is; in time, Lord Voldemort saw it too. He used that anger to his own purposes, and through careful compliments and attentions, win Snape to his own side -- in so doing, proving himself far crueler to Snape than any Gryffindor boy had ever been.
And so few at Hogwarts paid much attention when Harry Potter and Draco Malfoy began sniping at each other from almost the very first day. Gryffindors will be Gryffindors, Slytherins Slytherins. It all works out, in time, the teachers say. The Houses keep like with like so that, over time, the children find their uniqueness and individuality. A bit of in-house sparring is part and parcel of that.
But Dumbledore remembers, and watches, and wonders if this boyhood rivalry is the beginning of something far worse. Again.