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The study of hypersurfaces in a torus leads to the beautiful zoo of amoebasand their contours, whose possible configurations are seen from combinatorialdata. There is a deep connection to the logarithmic Gauss map and its criticalpoints. The theory has a lot of applications in many directions.  In this report we recall basic notions and results from the theory ofamoebas, show some connection to algebraic singularity theory and discuss someconsequences from the well known classification of singularities to thissubject. Moreover, we have tried to compute some examples using the computeralgebra system SINGULAR and discuss different possibilities and theireffectivity to compute the critical points. Here we meet an essential obstacle:Relevant examples need real or even rational solutions, which are found only bychance. We have tried to unify different views to that subject.

Discriminant and Singularities of Logarithmic Gauss Map, Examples and Application