  For the general obstacle problem, we prove by direct methods an epiperimetric
inequality at regular and singular points, thus answering a question of Weiss
(Invent. Math., 138 (1999), 23--50). In particular at singular points we
introduce a new tool, which we call logarithmic epiperimetric inequality, which
yields an explicit logarithmic modulus of continuity on the $C^1$ regularity of
the singular set, thus improving previous results of Caffarelli and Monneau.
