  The cosmic 21-cm signal will bring data-driven advances to studies of the
Cosmic Dawn (CD) and Epoch of Reionization (EoR). Radio telescopes such as the
SKA will eventually map the HI fluctuations over the first billion years - the
majority of our observable Universe. With such large data volumes, it becomes
increasingly important to develop "optimal" summary statistics, allowing us to
learn as much as possible about the CD and EoR. In this work we compare the
constraining power of several 21-cm summary statistics, using the determinant
of the Fisher information matrix, $\det F$. Since we do not have an established
"fiducial" model for the astrophysics of the first galaxies, we compute the
distribution of $\det F$ across the prior volume. Using a large database of
cosmic 21-cm lightcones that include realizations of telescope noise, we
compare the following summaries: (i) the spherically-averaged power spectrum
(1DPS), (ii) the cylindrically-averaged power spectrum (2DPS), (iii) the 2D
Wavelet scattering transform (WST), (iv) a recurrent neural network (RNN), (v)
an information-maximizing neural network (IMNN), and (vi) the combination of
2DPS and IMNN. Our best performing individual summary is the 2DPS, having
relatively high Fisher information throughout parameter space. Although capable
of achieving the highest Fisher information for some parameter choices, the
IMNN does not generalize well, resulting in a broad distribution. Our best
results are achieved with the concatenation of the 2DPS and IMNN. The
combination of only these two complimentary summaries reduces the recovered
parameter variances on average by factors of $\sim$6.5 - 9.5, compared with
using each summary independently. Finally, we point out that that the common
assumption of a constant covariance matrix when doing Fisher forecasts using
21-cm summaries can significantly underestimate parameter constraints.
