  The first electromagnetic signal from a supernova (SN) is released when the
shock crosses the progenitor surface. This shock breakout (SBO) emission
provides constraints on progenitor and explosion properties. Observationally,
SBOs appear as minute to hour-long extragalactic X-ray transients. They are
challenging to detect and only one SBO has been observed to date. Here, we
search the XMM-Newton archive and find twelve new SN SBO candidates. We
identify host galaxies to nine of these at estimated redshifts of 0.1-1. The
SBO candidates have energies of ${\sim}10^{46}$ erg, timescales of 30-3000 s,
and temperatures of 0.1-1 keV. They are all consistent with being SN SBOs, but
some may be misidentified Galactic foreground sources or other extragalactic
objects. SBOs from blue supergiants agree well with most of the candidates.
However, a few could be SBOs from Wolf-Rayet stars surrounded by dense
circumstellar media, whereas two are more naturally explained as SBOs from red
supergiants. The observations tentatively support non-spherical SBOs and are in
agreement with asymmetries predicted by recent three-dimensional SN explosion
simulations. eROSITA may detect ${\sim}$2 SBOs per year, which could be
detected in live analyses and promptly followed up.
