  Spontaneously broken, flavour-dependent, gauged $U(1)$ extensions of the
Standard Model (SM) have many phenomenological uses. We chart the space of
solutions to the gauge anomaly cancellation equations in such extensions, for
both the SM chiral fermion content and the SM plus (up to) three right-handed
neutrinos (SM$\nu_R$). Methods from Diophantine analysis allow us to
efficiently index the solutions arithmetically, and produce the complete
solution space in particular cases. In order to solve the general case, we
build a computer program which cycles through possible $U(1)$ charge
assignments, providing all solutions for charges up to some pre-defined maximum
absolute charge. Lists of anomaly-free $U(1)$ charge assignments result, which
corroborate the results of our Diophantine analysis. We make these lists, which
may be queried for further desirable properties, publicly available. This
previously uncharted space of anomaly-free charge assignments has been little
explored until now, paving the way for future model building and
phenomenological studies.
