  By using quasi--derivatives we develop a Fourier method for studying the
spectral gaps of one dimensional Schr\"odinger operators with periodic singular
potentials $v.$ Our results reveal a close relationship between smoothness of
potentials and spectral gap asymptotics under a priori assumption $v \in
H^{-1}_{loc} (\mathbb{R}).$ They extend and strengthen similar results proved
in the classical case $v \in L^2_{loc}(\mathbb{R}).$
