  We study the 1-D Schr\"odinger operators in Hilbert space $L^{2}(\mathbb{R})$
with real-valued Radon measure $q'(x)$, $q\in \mathrm{BV}_{loc}(\mathbb{R})$ as
potentials. New sufficient conditions for minimal operators to be bounded below
and selfadjoint are found. For such operators a criterion for the discreteness
of the spectrum is proved, which generalizes Molchanov's, Brinck's, and the
Albeverio-Kostenko-Malamud criteria. The quadratic forms corresponding to the
investigated operators are described.
