In several applications, such as wideband spectrum sensing for cognitive radio, only the power spectrum (a.k.a. the power spectral density) is of interest and there is no need to recover the original signal itself. In addition, high-rate analog-to-digital converters (ADCs) are too power hungry for direct wideband spectrum sensing. These two facts have motivated us to investigate compressive wideband power spectrum sensing, which consists of a compressive sampling procedure and a reconstruction method that is able to recover the unknown power spectrum of a wide-sense stationary signal from the obtained sub-Nyquist rate samples. It is different from spectrum blind sampling (SBS), which aims at reconstructing the original signal instead of the power spectrum. In this paper, a solution is first presented based on a periodic sampling procedure and a simple least-squares reconstruction method. We evaluate the reconstruction process both in the time and frequency domain. Then, we examine two possible implementations for the compressive sampling procedure, namely complex Gaussian sampling and multicoset sampling, although we mainly focus on the latter. A new type of multicoset sampling is introduced based on the so-called minimal sparse ruler problem. Next, we analyze the statistical properties of the estimated power spectrum. The computation of the mean and the covariance of the estimates allows us to calculate the analytical normalized mean squared error (NMSE) of the estimated power spectrum. Further, when the received signal is assumed to contain only circular complex zero-mean Gaussian i.i.d. noise, the computed mean and covariance can be used to derive a suitable detection threshold. Simulation results underline the promising performance of our proposed approach. Note that all benefits of our method arise without putting any sparsity constraints on the power spectrum.