  We construct two new one-parameter families of monotonicity formulas to study
the free boundary points in the lower dimensional obstacle problem. The first
one is a family of Weiss type formulas geared for points of any given
homogeneity and the second one is a family of Monneau type formulas suited for
the study of singular points. We show the uniqueness and continuous dependence
of the blowups at singular points of given homogeneity. This allows to prove a
structural theorem for the singular set.
  Our approach works both for zero and smooth non-zero lower dimensional
obstacles. The study in the latter case is based on a generalization of
Almgren's frequency formula, first established by Caffarelli, Salsa, and
Silvestre.
