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In the theory of bicomplex holomorphic functions, there is not concept of isolated singularities; that is, such functions do not have singularities just at a point like holomorphic functions in one complex variable. However, there are other type of singularities that behave similarly to the isolated singularities in one complex variable. In this work, we describe how they can be classified in such a way that it resembles the classification made for the complex analysis case. It turns out that to singularities there corresponds their orders which are hyperbolic numbers with integer components, not real integers. We give also the Residue Theorem in the bicomplex analysis setting.