Compressive sensing (CS) has received much attention in several fields such as digital image processing, wireless channel estimation, radar imaging, and cognitive radio (CR) communications. Out of these areas, this survey paper focuses on the application of CS in CR communications. Due to the under-utilization of the allocated radio spectrum, spectrum occupancy is usually sparse in different domains such as time, frequency, and space. Such a sparse nature of the spectrum occupancy has inspired the application of CS in CR communications. In this regard, several researchers have already applied the CS theory in various settings considering the sparsity in different domains. In this direction, this survey paper provides a detailed review of the state of the art related to the application of CS in CR communications. Starting with the basic principles and the main features of CS, it provides a classification of the main usage areas based on the radio parameter to be acquired by a wideband CR. Subsequently, we review the existing CS-related works applied to different categories such as wideband sensing, signal parameter estimation and radio environment map (REM) construction, highlighting the main benefits and the related issues. Furthermore, we present a generalized framework for constructing the REM in compressive settings. Finally, we conclude this survey paper with some suggested open research challenges and future directions.