In this paper, a new approach for the analysis of three-phase electrical signals is considered. While most of the existing techniques are based on fixed transforms such as the Clarke transform, this paper investigates the use of a data-driven approach called the singular value decomposition (SVD). As compared to other transforms, this paper shows that the SVD has the distinct advantage of clearly separating the contributions of the phasor configuration and signal instantaneous parameters. Under additive white Gaussian noise, this paper also describes several algorithms based on the SVD for signal monitoring. The first algorithm can detect unbalanced systems and classify them into two categories: 1) system with an off-nominal subspace and 2) systems with ellipticity. The second algorithm can estimate the angular frequency based on the periodogram of the right singular vectors. As compared to other existing approaches, simulations show that the proposed techniques give a good compromise between computational complexity and statistical performance.