There are one hundred and forty-two staircases at Hogwarts:
wide, sweeping ones; narrow, rickety ones; some that led somewhere different on a Friday; some with a vanishing step halfway up that you had to remember to jump (PS8).
One of them, located on the third floor, is described as a swivelling staircase (OP32).
The staircase with the vanishing step extends from the second to the fourth floor. It had frequently caught Neville, but Harry himself found himself stuck while wandering around under his invisibility cloak and focusing on the Marauder’s Map rather than where he was going (GF25). That staircase is located along a popular shortcut up to Gryffindor Tower. The bottom of the staircase is concealed behind a tapestry on the second floor (HBP14, HBP28).
The iconic visual of the staircases swinging around is purely a film invention. The staircases in the books, with the single exception of the swivelling staircase on the third floor, are not described as moving around at all. Pottermore's depiction matches that of the books: