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This is the first of a series of papers in which we systematically use singularity theory to study four dimensional N= 2 superconformal field theories. Our main focus in this paper is to identify what kind of singularity is needed to define a SCFT. The constraint for a hypersurface singularity has been found by Sharpere and Vafa, and here the complete set of solutions are listed using a related mathematical result of Stephen ST Yau and Yu. We also study other type of singularities such as the complete intersection, quotient of hypersurface singularity by a finite group and non-isolated singularity. We finally conjecture that any three dimensional rational Gorenstein graded isolated singularity should define a N= 2 SCFT. We explain how to extract various interesting physical quantities such as Seiberg-Witten geometry, central charges, exact marginal deformations, BPS quiver, RG flow trajectory, etc from the properties of singularity.

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