  New physics can manifest itself by an appreciable increase of the decay rate
of top quarks in rare flavour-changing final states. Exploiting the large top
quark production rate at the LHC, we bound four-fermion operators contributing
to non-resonant $t\rightarrow \ell^+\ell^- j$ using different signal regions of
the latest LHC searches for $t\rightarrow Zj$. We also provide prospects for
the high-luminosity LHC to test these as well as four-fermion operators
contributing to $t\rightarrow b\overline{b}j$, based on improved analysis
strategies of existing searches. We single out all weakly-coupled ultraviolet
completions inducing such contact interactions at tree level and translate the
previous bounds to the parameter space of specific complete models. Being above
the TeV, LHC bounds from rare top decays improve over those from flavour
physics, electroweak precision data and other LHC searches in several cases.
