We generalize the analysis of [?] and develop a singular pseudodifferential calculus. The symbols that we consider do not satisfy the standard decay with respect to the frequency variables. While in [?], the remainders in the symbolic calculus were seen to be merely bounded operators on $L^2$, whose norm was measured with respect to some small parameter, we show here a smoothing property for the remainders. Due to a nonstandard decay in the frequency variables, the smoothing effect takes place in a scale of anisotropic, and singular, Sobolev spaces. Our analysis allows to extend the results of [?] on the existence of highly oscillatory solutions to nonlinear hyperbolic problems by dropping the compact support condition on the data. The results are also used in our companion works [?] to justify nonlinear geometric optics with boundary amplification, which corresponds to a more singular regime than the one considered in [?]. The analysis is carried out with either an additional real or periodic variable in order to cover problems for pulses or wavetrains in nonlinear geometric optics.