In this memory we follow the geometric approach of Casas’ boof [1] for studying of the singularities of plane germs of curves, which updates Enriques’ works to modern standards and reviews the modern development of the theorey from the point of view of infinitely near points. This memory has a two sided goal: on one hand, we want to acquire skills with the tools and concepts of the singularity theory and the valuative theory, both the classical ones and the more recent ones. On the other hand, we want to study in depth the different implicit concepts and notions involved in the Favre and Jonsson’s new approach, such as the ultrametric space structure of the set of irreducible germs of plane curves and the tree structure of the valuations.