Some of the cluster extensions of the coherent potential approximation (CPA) based on the effective medium theory have been critically studied with respect to the decoupling schemes involved in them. Their computational tractability has been examined and it has been found that theself-consistent calculations in three-dimensional systems are immensely difficult to perform. A self-consistent calculation has been reported for simple cubic lattices with diagonal and off-diagonal disorder using a pair-CPA method. A significant finding of the paper is that it has been shown thatnon-analyticities are a general feature of extensions of CPA within multiple scattering framework. The non-analyticities were reported several times but a general proof of their existence was not noticed. It was also believed that the so-called molecular—CPA is analytic, this has been shown to be wrong here. The density of states results with off-diagonal randomness have been qualitatively understood to yield some information about the influence of off-diagonal randomness on Anderson localisation of an electron.