There is now convincing evidence that the Standard Model of electroweak and strong interactions is not the end of the story but only a low energy effective theory. In particular, new flavour physics is required to explain the fermion mass spectrum. Most of the proposed extensions of the Standard Model fail to meet this criterion. We may hope that the LHC or some future colliders could help to clarify the situation by discovering new particles or spotting some unexpected events. In the meantime, more precise measurements of masses and mixing parameters could also play an important role. In this work, we do not aim at finding a new mechanism that could explain this spectrum, but we rather assume that fermion masses and mixings are calculable in a yet-to-be-found more fundamental theory. Our goal is to glean as much information as possible from the observed fermion masses and mixings in order to find some hidden structures that could significantly lower the number of free parameters and help us to get some clues about what could be this fundamental theory. We analyse first the various parametrizations of the flavour mixing and single out a specific decomposition. The parameters of this decomposition can be independently and accurately computed if we impose some simple textures to the Yukawa couplings. We propose then a straightforward combination of these interesting textures which reproduces quite well the observed quark flavour mixing. We study then the properties of a successful mass relation for the charged leptons. We propose some generalizations of this relation in order to be valid also for the neutrinos and the quarks. One of them successfully combines the masses and mixings while another one describes the lepton masses via an accurate geometric description. Hopefully, these two studies lead to similar conclusions and allow us to speculate on some interesting properties for new flavour physics.