This survey provides the reader with the open literature on the detailed operational scenarios of Target Cell Search (TCS) in both 3G and 4G systems, with a particular emphasis on the DownLink (DL). There is no completely latest survey of the TCS scenarios. Therefore, we examine the TCS-involved solutions which have been chosen for Wideband-Code Division Multiple Access (W-CDMA), Time Division-Synchronous Code Division Multiple Access (TD-SCDMA), Long-Term Evolution Frequency Division Duplex (LTE-FDD), and Time-Division Long-Term Evolution (TD-LTE). Especially for TD-SCDMA, there is no concrete literature demystifying the details of TCS scenario. Accordingly, this survey aims for repairing the associated crack in the literature. Moreover, as a recent evolution from the operational systems, we also focus on the TCS scenario which can be exploited in New Radio (NR) systems operating at both sub-6 GHz and millimeter-Wave (mmW) spectrum bands. A taxonomy diagram is also provided to classify relevant target cell search scenarios in 3G, 4G, and 5G as well as followed by those in inter-Radio Access Technologies (RATs). In a nutshell, our endeavors definitize the first comprehensive survey on the TCS progress tailored to the evolution of commercial mobile communication systems over three decades. It is worth mentioning that this survey is also informative for engineers and researchers to design practical mobile station modem.