  The distribution of matter fluctuations in our universe is key for
understanding the nature of dark matter and the physics of the early cosmos.
Different observables have been able to map this distribution at large scales,
corresponding to wavenumbers $k\lesssim 10$ Mpc$^{-1}$, but smaller scales
remain much less constrained. The 21-cm line is a promising tracer of early
stellar formation, which took place in small haloes (with masses $M\sim
10^6-10^8M_\odot$), formed out of matter overdensities with wavenumbers as
large as $k\approx100$ Mpc$^{-1}$. Here we forecast how well both the 21-cm
global signal, and its fluctuations, could probe the matter power spectrum
during cosmic dawn ($z=12-25$). We find that the long-wavelength modes (with
$k\lesssim40$ Mpc$^{-1}$) are highly degenerate with astrophysical parameters,
whereas the modes with $k= (40-80)$ Mpc$^{-1}$ are more readily observable.
This is further illustrated in terms of the principal components of the matter
power spectrum, which peak at $k\sim 50$ Mpc$^{-1}$ both for a typical
experiment measuring the 21-cm global signal and its fluctuations. We find
that, imposing broad priors on astrophysical parameters, a global-signal
experiment can measure the amplitude of the matter power spectrum integrated
over $k= (40-80)$ Mpc$^{-1}$ with a precision of tens of percent. A fluctuation
experiment, on the other hand, can constrain the power spectrum to a similar
accuracy over both the $k=(40-60)$ Mpc$^{-1}$ and $(60-80)$ Mpc$^{-1}$ ranges
even without astrophysical priors. The constraints outlined in this work would
be able to test the behavior of dark matter at the smallest scales yet
measured, for instance probing warm-dark matter masses up to $m_{\rm WDM}=8$
keV for the global signal and $14$ keV for the 21-cm fluctuations. This could
shed light on the nature of dark matter beyond the reach of other cosmic
probes.
