Sleep-disordered breathing (SDB) is a highly prevalent condition associated with many adverse health problems. As the current means of diagnosis (polysomnography) is obtrusive and ill-suited for mass screening of the population, we explore a non-contact, automatic approach that uses acoustics-based methods. We present a method for automatically classifying breathing sounds produced during sleep. We compare the performance of several acoustic feature representations for detecting diagnostically-relevant sleep breathing events to predict overall SDB severity. Our subject-independent method tracks rest in the breathing cycle with 84-87% accuracy, and predicts SDB severity at a level similar to polysomnography.