A sample of over 12,487 asteroid proper element triplets, computed by A. Milani and Z. Knežević (1994, Icarus 107, 219-254), has been searched for statistically significant and robust families by both the hierarchical clustering (HCM) and the wavelet analysis (WAM) automated techniques. The current sample includes almost 8000 unnumbered objects with relatively well-determined orbits, which allow us to extend the previous family classifications to much smaller asteroid sizes. With both clustering methods about one-third of the whole asteroid population is found to belong to groupings of varying significance and robustness. Nearly all the families identified in previous searches by Ph. Bendjoya (1993, Astron. Astrophys. Suppl. Ser. 102, 25-55) and V. Zappalà et al. (1994, Astron. J. 107, 772-801) are confirmed. Many more groupings are now recognized having a small number of members and/or a large percentage of unnumbered ones, particularly when the WAM technique is applied to asteroids located in the middle region of the main belt, between the 3:1 and the 5:2 Kirkwood gaps. Most of these new small families appear to derive from the (possibly recent) break-up of small parent asteroids, <50 km across. Some are very compact, suggesting ejection speeds not much in excess of the parent's escape velocity. Some fairly populous families can be readily interpreted as the outcome of giant cratering events, including those associated with (4) Vesta and (10) Hygiea. Others (e.g., Meliboea in the outer belt) are so depleted in small members that probably an old age is implied. Several populous families, and in particular those named "clans" (e.g., Eunomia, Vesta, Nysa), show a complex and size-dependent internal structure, suggesting a complex collisional history (i.e., multiple successive disruptive events). A problem with family searches applied to very numerous asteroid samples, such as the current one, is that neighboring, physically distinct groupings often partially overlap each other, and thus merge according to the clustering techniques.