This thesis is devoted to the study of singularities of holomorphic maps: their geometry, as well as cohomological and K-theoretic invariants, their properties and computational strategies. One of the main problems in global singularity theory is how to compute Thom polynomials. I show how the two modern methods of computation can be combined in a different computational approach and give examples of computation. A recent development in global singularity theory is the introduction of the K-theoretic invariants of singularity loci. One can define a K-theoretic invariant of an affine variety in two different ways. I prove that even for A2 singularity loci, in the general case, the two invariants are different, and therefore, the A2-loci may have singularities worse than rational. However, in the case of relative codimension 0, the two invariants coincide, and thus the A2-loci have rational singularities.