Near-Field Acoustical Holography (NAH) is an inverse process in which sound pressure measurements made in the near-field of an unknown sound source are used to reconstruct the sound field so that source distributions can be clearly identified. NAH was originally based on performing spatial transforms of arrays of measured pressures and then processing the data in the wavenumber domain, a procedure that entailed the use of very large microphone arrays to avoid spatial truncation effects. Over the last twenty years, a number of different NAH methods have been proposed that can reduce or avoid spatial truncation issues: for example, Statistically Optimized Near-Field Acoustical Holography (SONAH), various Equivalent Source Methods (ESM), etc. Then, more recently, with the motivation of facilitating the measurement process, the principles of Compressive Sensing (CS) have been introduced in several studies to allow sound fields to be reconstructed based on a relatively small number of microphone measurements (thus making holographic measurements more practical and inexpensive), and these studies have shown promising results when used to identify sound source locations. In the present work, the ESM based on an assumed monopole source distribution was the NAH method studied, and the inverse problem that is required to identify the equivalent source strengths was solved by using two different CS algorithms: Wideband Acoustical Holography (WBH) and l1-norm convex optimization (l1- CVX). Several different source types were chosen to test the reconstruction capabilities of the two algorithms: in particular, concentrated point source distributions, and spatially-extended sources such as baffled plate vibration. The strengths and weaknesses of the two CS algorithms have been identified with reference to results obtained by using SONAH.