The exponential growth of astronomical data collected by both ground based and space borne instruments has fostered the growth of Astroinformatics: a new discipline laying at the intersection between astronomy, applied computer science, and information and computation (ICT) technologies. At the very heart of Astroinformatics is a complex set of methodologies usually called Data Mining (DM) or Knowledge Discovery in Data Bases (KDD). In the astronomical domain, DM/KDD are still in a very early usage stage, even though new methods and tools are being continuously deployed in order to cope with the Massive Data Sets (MDS) that can only grow in the future. In this paper, we briefly outline some general problems encountered when applying DM/KDD methods to astrophysical problems, and describe the DAME (DAta Mining & Exploration) web application. While specifically tailored to work on MDS, DAME can be effectively applied also to smaller data sets. As an illustration, we describe two application of DAME to two different problems: the identification of candidate globular clusters in external galaxies, and the classification of active galactic nuclei (AGN). We believe that tools and services of this nature will become increasingly necessary for the data-intensive astronomy (and indeed all sciences) in the 21st century.