  Spectroscopic surveys of massive galaxy clusters reveal the properties of
faint background galaxies, thanks to the magnification provided by strong
gravitational lensing. We present a systematic analysis of
integral-field-spectroscopy observations of 12 massive clusters, conducted with
the Multi Unit Spectroscopic Explorer (MUSE). All data were taken under very
good seeing conditions (0.6") in effective exposure times between two and 15
hrs per pointing, for a total of 125 hrs. Our observations cover a total solid
angle of ~23 arcmin$^2$ in the direction of clusters, many of which were
previously studied by the MACS, Frontier Fields, GLASS and CLASH programs. The
achieved emission line detection limit at 5$\sigma$ for a point source varies
between (0.77--1.5)$\times$10$^{-18}$ erg\,s$^{-1}$\,cm$^{-2}$ at 7000\AA. We
present our developed strategy to reduce these observational data, detect
sources and determine their redshifts. We construct robust mass models for each
cluster to further confirm our redshift measurements using strong-lensing
constraints, and identify a total of 312 strongly lensed sources producing 939
multiple images. The final redshift catalogs contain more than 3300 robust
redshifts, of which 40\% are for cluster members and $\sim$30\% for lensed
Lyman-$\alpha$ emitters. 14\% of all sources are line emitters not seen in the
available HST images, even at the depth of the FFs ($\sim29$ AB). We find that
the magnification distribution of the lensed sources in the high-magnification
regime ($\mu{=}$ 2--25) follows the theoretical expectation of
$N(z)\propto\mu^{-2}$. The quality of this dataset, number of lensed sources,
and number of strong-lensing constraints enables detailed studies of the
physical properties of both the lensing cluster and the background galaxies.
The full data products from this work are made available to the community.
[abridged]
