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We present new results on the time-dependent correlation functions Ξn(t)=4⟨Sξ0(t)Sξn⟩, ξ=x,y, at zero temperature of the one-dimensional S=12 isotropic XY model (h=γ=0) and of the transverse Ising (TI) model at the critical magnetic field (h=γ=1). Both models are characterized by special cases of the Hamiltonian H=−JΣl[(1+γ)SxlSxl+1+(1−γ)SylSyl+1+hSz1]. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions Ξ0(0) and on the singularities of their frequency-dependent Fourier transforms Φξξ0(ω). We have also determined the latter functions by high-precision numerical calculations. The functions Φξξ0(ω), ξ=x,y, have singularities at the infinite sequence of frequencies ω=mω0, m=0,1,2,3,…, where ω0=J for the XY model and ω0=2J for the TI model. In the TI case, the leading singularities in φxx0(ω) are alternately one-sided and two-sided power-law singularities, the first two of which (at ω=0,2J) are divergent. The dominant singularities in the XY case are alternate one-sided power laws and two-sided power laws with logarithmic corrections, the first two of which (at ω=0,J) are divergent. The singularities at higher frequencies in both models are finite and become increasingly weaker. We point out that the nonanalyticities at ω≠0 are intrinsic features of the discrete quantum chain and have therefore not been found in the context of a continuum analysis (Luttinger model). At least the most prominent features of our new results should be observable in low-temperature dynamical experiments on quasi-one-dimensional compounds such as the XY-like substances Cs2CoCl4 and PrCl3 and the S=12 Ising-like substance CsCoCl3·2H2O.