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Cognitive radios sense the radio spectrum in order to find underutilized spectrum and then exploit it in an agile manner. Spectrum sensing has to be performed reliably in challenging propagation environments characterized by shadowing and fading effects as well as heavy-tailed noise distributions. In this paper, a robust computationally efficient nonparametric cyclic correlation estimator based on the multivariate (spatial) sign function is proposed. Nonparametric statistics provide additional robustness against heavy-tailed noise and when the noise statistics are not fully known. Asymptotic distribution of the spatial sign cyclic correlation estimator under the null hypothesis is established. Tests using constraint on false alarm rate are derived based on the estimated spatial sign cyclic correlation for single-user and collaborative spectrum sensing by multiple secondary users. Theoretical justification for detecting cyclostationary signals using the spatial sign cyclic correlation is provided. A sequential detection scheme for reducing the average detection time is proposed. Simulation experiments and theoretical results comparing the proposed method with cyclostationary spectrum sensing methods employing the conventional cyclic correlation estimator are presented. Simulations demonstrate the reliable and highly robust performance of the proposed nonparametric spectrum sensing method in both Gaussian and non-Gaussian noise environments.