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We analyze the formation of the redshifted hyperfine structure line 21-cm of hydrogen atoms in Dark Ages at 30 ≤ z ≤ 300 in different cosmologies. To study its dependence on the values of cosmological parameters and physical conditions in the intergalactic medium, the evolution of the global (sky-averaged) differential brightness temperature in this line was computed in standard and non-standard cosmological models with different parameters. The standard ΛCDM model with post-Planck parameters predicts the following value of the differential brightness temperature in the center of the absorption line: δTbr ≈ 35 mK at z ≈ 87. The frequency of the line in the absorption maximum is 16 MHz, the effective half-width of the line is 17 MHz. The depth of the line is moderately sensitive to Ωb and H0, weakly sensitive to Ωdm, and insensitive to other parameters of the standard ΛCDM model. But the line is very sensitive to additional mechanisms of heating or cooling of baryonic matter during the Dark Ages, so it can be a good test of non-standard cosmological models. In the models featuring decaying and self-annihilating dark matter, along with a primordial stochastic magnetic field, the temperature of baryonic matter in this period increases with higher densities of these dark matter components and stronger magnetic field strengths. The absorption line gradually becomes shallower, and eventually disappears, transitioning into emission when the values of the component parameters are lower than their upper limits, as determined by current observational data. Estimates show that such spectral features may be detected by radio telescopes in the decameter wavelength range in the near future. [ABSTRACT FROM AUTHOR]