Classify isolated singularities in three examples — with one unresolved extractor defect

Find and classify all isolated singularities, giving pole orders and residues where applicable, for \[ \frac{\cos z-1}{z^2}, \qquad \frac{e^{1/z^2}}{z-1}, \] and for the third function, which the deterministic extraction renders as \[ \frac{\sqrt{\overline z}}{z-1}. \] The third formula is not holomorphic on a punctured neighborhood and therefore cannot literally fit the requested isolated-singularity classification; its source formula remains unresolved.