  Leptoquarks have recently received much attention especially because they may
provide an explanation to the $R_{D^{(*)}}$ and $R_{K^{(*)}}$ anomalies in rare
$B$ meson decays. In a previous paper we proposed a systematic search strategy
for all possible leptoquark flavors by focusing on leptoquark pair production.
In this paper, we extend this strategy to large (order unity) leptoquark
couplings which offer new search opportunities: single leptoquark production
and $t$-channel leptoquark exchange with dilepton final states. We discuss the
unique features of the different search channels and show that they cover
complementary regions of parameter space. We collect and update all currently
available bounds for the different flavor final states from LHC searches and
from atomic parity violation measurements. As an application of our analysis,
we find that current limits do not exclude the leptoquark explanation of the
$B$ physics anomalies but that the high luminosity run of the LHC will reach
the most interesting parameter space.
