An effective solution of the problem of analytic classification of plane branches is given. A finite stratification of any given equisingularity class of plane branches is determined and normal forms for each stratum are exhibited in such a way that two branches in normal form are easily recognized to be analytically equivalent or not. In this way, we solve the main problems proposed by O. Zariski (Le problème des modules pour les branches planes.

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We perform the analytic classification of plane branches of multiplicity less or equal than four. This is achieved by computing a Standard basis for the modules of Kähler differentials of such branches by means of the algorithm we developed in [9] and then applying the classification method for plane branches presented in [10].