  We employ a bottom-up and model-independent technique to search for
non-Abelian discrete flavour symmetries capable of predicting viable CKM and
PMNS matrices alongside of special patterns of leptoquark couplings. In
particular, we analyze patterns derived when an ultra-violet flavour theory is
assumed to break to global Abelian symmetries in Standard Model fermion masses
and new Yukawa-like terms sourced by the leptoquark representation. The
phenomenology of different classes of these `simplified models' can be explored
without reference to explicit model-building assumptions, e.g. the nature of
flavour symmetry breaking or any additional field content associated to it, and
are also capable of explaining hints of lepton non-universality in
$\mathcal{R}_{K^{(\star)}}$. Assuming experimentally interesting CKM and PMNS
matrix elements, our algorithm finds an abundance of predictive non-Abelian
flavour groups and therefore provides promising directions for future model
building in the flavoured leptoquark space, regardless of whether the anomalous
$\mathcal{R}_{K^{(\star)}}$ measurements withstand further experimental
scrutiny.
