  We discuss the local behaviour of vector fields in the plane $\R^2$ around a
regular singular point, using recently introduced reduced normal forms, i.e.
Poincar\'e and Lie renormalized forms [{\it Lett. Math. Phys.} {\bf 42} (1997),
103-114; {\it Ann. Inst. H. Poincar\'e (Phys. Theo.)} {\bf 70} (1999), 461-514;
{\it Lett. Math. Phys.} {\bf 57} (2001), 41-60]. We give a complete
classification, and provide explicit formulas, using Poincar\'e renormalized
forms for non-degenerate cases, and Lie ones for certain degenerate cases. Both
schemes are completely algorithmic, prove to be easy to implement, and only
require to solve linear equations.
