In the context of a renormalizable supersymmetric SO(10) Grand Unified Theory, we consider the fermion mass matrices generated by the Yukawa couplings to a $\mathbf{10} \oplus \mathbf{120} \oplus \bar{\mathbf{126}}$ representation of scalars. We perform a complete investigation of the possibilities of imposing flavour symmetries in this scenario; the purpose is to reduce the number of Yukawa coupling constants in order to identify potentially predictive models. We have found that there are only 14 inequivalent cases of Yukawa coupling matrices, out of which 13 cases are generated by $Z_n$ symmetries, with suitable $n$, and one case is generated by a $Z_2 \times Z_2$ symmetry. A numerical analysis of the 14 cases reveals that only two of them---dubbed A and B in the present paper---allow good fits to the experimentally known fermion masses and mixings.

36 pages, no figures, revised fits using newer data, added fit for case A, added references, new appendices concerning the SO(10) scalar potential and inequalities for the vacuum expectation values, conclusions unchanged; some minor changes, matches published version