Spectrum sensing is a crucial component of opportunistic spectrum access schemes, which aim at improving spectrum utilization by allowing for the reuse of idle licensed spectrum. Sensing a spectral band before using it makes sure the legitimate users are not disturbed. Since information about these users' signals is not necessarily available, the sensor should be able to conduct so-called blind spectrum sensing. Historically, this has not been a feature of cyclostationarity-based algorithms. Indeed, in many application scenarios the information required for traditional cyclostationarity detection might not be available, hindering its practical applicability. In this work we propose two new cyclostationary spectrum sensing algorithms that make use of the inherent sparsity of the cyclic autocorrelation to make blind operation possible. Along with utilizing sparse recovery methods for estimating the cyclic autocorrelation, we take further advantage of its structure by introducing joint sparsity as well as general structure dictionaries into the recovery process. Furthermore, we extend a statistical test for cyclostationarity to accommodate sparse cyclic spectra. Our numerical results demonstrate that the new methods achieve a near constant false alarm rate behavior in contrast to earlier approaches from the literature.