  Given the hints of lepton-flavour non-universality in $B$-meson decays,
leptoquarks (LQs) are enjoying a renaissance. We propose novel Large Hadron
Collider (LHC) searches for such hypothetical states that do not rely on strong
production only, but can also receive important contributions from quark-lepton
annihilation. For the cases of a resonant signal involving a bottom quark and a
tau lepton ($b + \tau$), a top quark and missing transverse energy ($E_{T}^{\rm
miss}$) and light-flavour jets plus $E_{T}^{\rm miss}$, we develop realistic
analysis strategies and provide detailed evaluations of the achievable
sensitivities for the corresponding LQ signatures at future LHC runs. Our
analyses allow us to derive a series of stringent constraints on the masses and
couplings of third-generation singlet vector LQs, showing that at LHC Run III
and the high-luminosity LHC the proposed search channels can probe interesting
parts of the LQ parameter space addressing the $B$-physics anomalies. In view
of the reach of the proposed $b + \tau$ signature, we recommend that dedicated
resonance searches for this final state should be added to the exotics search
canon of both ATLAS and CMS.
