"Harry glanced down at his broom. It was old and some of the twigs stuck out at odd angles." (PS9)
Broomsticks and broom sports hold a special place in the hearts of Hogwarts students. Quidditch matches are routinely attended by nearly every student and teacher in the school, and star fliers have near-celebrity status among the student body.
Brooms, even those owned by students, are normally stored in a broomshed by the Quidditch pitch (PS13), although when Harry gets a Firebolt he keeps it in his dormitory out of concern for its safety (PA15). This is probably smart, as the broomshed can get quite cold - Hagrid is seen outside on one cold fall morning defrosting the brooms from the shed (PS11). Hogwarts owns a number of brooms, but they are old and don't fly especially well; Harry dreads the thought of playing a Quidditch match on one after his broom is destroyed (PA10). These brooms are used primarily for flying lessons, which first years take with Madame Hooch (PS9), though most of them can't fly very well (HBP11).