We examine new aspects of leptoquark (LQ) phenomenology using effective field
theory (EFT). We construct a complete set of leading effective operators
involving SU(2) singlets scalar LQ and the SM fields up to dimension six. We
show that, while the renormalizable LQ-lepton-quark interaction Lagrangian can
address the persistent hints for physics beyond the Standard Model in the
B-decays $\bar B \to D^{(*)} \tau \bar\nu$, $\bar B \to \bar K \ell^+ \ell^-$
and in the measured anomalous magnetic moment of the muon, the LQ higher
dimensional effective operators may lead to new interesting effects associated
with lepton number violation. These include the generation of one-loop sub-eV
Majorana neutrino masses, mediation of neutrinoless double-$\beta$ decay and
novel LQ collider signals. For the latter, we focus on 3rd generation LQ
($\phi_3$) in a framework with an approximate $Z_3$ generation symmetry, and
show that one class of the dimension five LQ operators may give rise to a
striking asymmetric same-charge $\phi_3 \phi_3$ pair-production signal, which
leads to low background same-sign leptons signals at the LHC. For example, with
$M_{\phi_3} \sim 1$ TeV and a new physics scale of $\Lambda \sim 5$ TeV, we
expect about $5000$ positively charged $\tau^+ \tau^+$ events via $pp \to
\phi_3 \phi_3 \to \tau^+ \tau^+ + 2 \cdot j_b$ ($j_b$=b-jet) at the 13 TeV LHC
with an integrated luminosity of 300 fb$^{-1}$. It is interesting to note that,
in the LQ EFT framework, the expected same-sign lepton signals have a rate
which is several times larger than the QCD LQ-mediated opposite-sign leptons
signals, $gg,q \bar q \to \phi_3 \phi_3^* \to \ell^+ \ell^- +X$. We also
consider the same-sign charged lepton signals in the LQ EFT framework at higher
energy hadron colliders such as a 27 TeV HE-LHC and a 100 TeV FCC-hh.