We present here 21 cm predictions from high dynamic range simulations for a range of reionization histories that have been tested against available Ly $\alpha$ and cosmic microwave background (CMB) data. We assess the observability of the predicted spatial 21 cm fluctuations by ongoing and upcoming experiments in the late stages of reionization in the limit in which the hydrogen spin temperature is significantly larger than the CMB temperature. Models consistent with the available Ly $\alpha$ data and CMB measurement of the Thomson optical depth predict typical values of 10–20 mK$2$ for the variance of the 21 cm brightness temperature at redshifts $z$ = 7–10 at scales accessible to ongoing and upcoming experiments ($k$ $\lesssim$ 1 cMpc$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}h$). This is within a factor of a few magnitude of the sensitivity claimed to have been already reached by ongoing experiments in the signal rms value. Our different models for the reionization history make markedly different predictions for the redshift evolution and thus frequency dependence of the 21 cm power spectrum and should be easily discernible by Low-Frequency Array (and later Hydrogen Epoch of Reionization Array and Square Kilometre Array1) at their design sensitivity. Our simulations have sufficient resolution to assess the effect of high-density Lyman limit systems that can self-shield against ionizing radiation and stay 21 cm bright even if the hydrogen in their surroundings is highly ionized. Our simulations predict that including the effect of the self-shielded gas in highly ionized regions reduces the large-scale 21 cm power by about 30 per cent.