The growing demand for nano-sized and efficient semiconductors leading to a technological revolution in quantum technology has unleashed into the development of new types of compound semiconductors of nano-scale. However, the physical phenomena limiting their efficiency requires more study into the charge transportation phenomena. The study of instabilities like drift instability and modulation instability of the waves excited in a semiconductor plasma due to various force field configurations with the inclusion of quantum effects has gained recent attention, as the solid-state plasma in a semiconductor satisfies the condition for quantum plasma. The different types of instabilities, its mechanism, and effect in a semiconductor quantum plasma, have been studied in detail and presented in this article. Most of the works that has been carried out to study the instabilities have used the quantum hydrodynamic (QHD) model. The important quantum effects that are highlighted in most of the work includes the Bohm potential, exchange potential, and Fermi degenerate pressure in the nano-sized quantum semiconductors. This review work may be relevant to all who wants to have an insight on various instabilities in semiconductor quantum plasma.