We present sublinear algorithms -algorithms that use significantly less resources than needed to store or process the entire input stream - for discovering representative trends in data streams in the form of periodicities. Our algorithms involve sampling Õ(√n) positions. and thus they scan not the entire data stream but merely a sublinear sample thereof. Alternately, our algorithms may be thought of as working on streaming inputs where each data item is seen once, but we store only a sublinear - Õ(√n) - size sample from which we can identify periodicities. In this work we present a variety of definitions of periodicities of a given stream, present sublinear sampling algorithms for discovering them, and prove that the algorithms meet our specifications and guarantees. No previously known results can provide such guarantees for finding any such periodic trends. We also investigate the relationships between these different definitions of periodicity.