This paper presents two new algorithms for wideband spectrum sensing at sub-Nyquist sampling rates, for both single nodes and cooperative multiple nodes. In single-node spectrum sensing, a two-phase spectrum sensing algorithm based on compressive sensing is proposed to reduce the computational complexity and improve the robustness at secondary users (SUs). In the cooperative multiple nodes case, the signals received at SUs exhibit a sparsity property that yields a low-rank matrix of compressed measurements at the fusion center. This therefore leads to a two-phase cooperative spectrum sensing algorithm for cooperative multiple SUs based on low-rank matrix completion. In addition, the two proposed spectrum sensing algorithms are evaluated on the TV white space (TVWS), in which pioneering work aimed at enabling dynamic spectrum access into practice has been promoted by both the Federal Communications Commission and the U.K. Office of Communications. The proposed algorithms are tested on the real-time signals after they have been validated by the simulated signals in TVWS. The numerical results show that our proposed algorithms are more robust to channel noise and have lower computational complexity than the state-of-the-art algorithms.