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Compressive sensing (CS) has recently emerged as an extremely efficient technology of the wideband spectrum sensing. In compressive spectrum sensing (CSS), it is necessary to know the sparsity or the noise information in advance for reliable reconstruction. However, such information is usually absent in practical applications. In this paper, we propose a blind-block orthogonal least squares-based compressive spectrum sensing (B-BOLS-CSS) algorithm, which utilizes a novel blind stopping rule to cut the cords to these prior information. Specifically, we first present both the noiseless and noisy recovery guarantees for the BOLS algorithm based on the mutual incoherence property (MIP). Motivated by them, we then formulate the blind stopping rule, which exploits an $\ell_{2,\infty}$ sufficient statistic to blindly test the support atoms in the remaining measurement matrix. We further evaluate the theoretical performance analysis of the holistic B-BOLS-CSS algorithm by developing a lower bound of the signal-to-noise ratio (SNR) to ensure that the probability of exact recovery is no lower than a given threshold. Simulations not only demonstrate the improvement of our derived theoretical results, but also illustrate that B-BOLS-CSS works well in both low and high SNR environments.