In this paper, the locally optimum detector for complex-valued cyclostationary signals is derived. Locally optimum detection is the optimum detection scheme for weak signal detection. However, the derived locally optimum detector requires synchronization to the signal to be detected and knowledge of the noise probability density function (pdf) and its parameters. Hence, suboptimum single-cycle detectors of conjugate and nonconjugate cyclostationarity that do not require complete synchronization are considered as well. Moreover, simulation results show that locally optimum and single-cycle detectors designed assuming a complex Cauchy noise model have highly reliable or even optimal performance in heavy-tailed noise and close-to-optimal performance in Gaussian noise.