  A complete study of local singularities of rank two $\mathcal{N}=2$ Coulomb
branch geometry is given. Low energy theory associated with the local
singularity is identified: it can be superconformal field theory (SCFT), or IR
free gauge theory, or the combination of them. Various invariants for local
singularity are also listed which are essential for the study of global Coulomb
branch. As a first application, global Coulomb branch with only simplest local
singularities in the bulk are given for 4d theories (including SCFTs and
asymptotical free theories), 5d KK theories, and 6d KK theories; those examples
appear to cover all the findings in the literature and suggest there are more
possibilities. More general global Coulomb branch geometry would be discussed
in the sequel of this paper.
