Singularity Analysis refers to the process of evaluating the local singularity exponent of a given signal at all points of its domain. The great potential of using singularity analysis for the description and understanding of multiscale systems have been repeatedly evidenced during the recent years. Simple algorithms for singularity analysis work well from the point of view of the statistical characterization of the system, but their quality, processing speed and overall performance are rather mediocre when more geometric, specific applications are envisaged. In this paper, we present a new algorithm which provides a fast characterization of singularity exponents acording to a central property observed in many multifractal systems: reconstructibility. Our singularity analysis scheme is based on a predictability measure, which provides information about the extent to which the signal at a given point can be reconstructed knowing its value over a neighborhood around that point. The singularity exponents so obtained attain very good spatial localization; more importantly, the most singular component associated to the obtaind exponents allow to reconstruct the signal to high accuracy. We finally provide some applications of our method for various problems in image procesing and series analysis.