Short-reach transmission systems traditionally utilize intensity modulation (IM) at the transmitter and direct detection (DD) at the receiver due to their cost-effectiveness, small footprint, and low power consumption. However, with the exponential increase in bandwidth demand, coherent optical communication systems have become necessary for long-haul distances, requiring application-specific integrated circuits (ASIC) and advanced digital signal processing (DSP) algorithms coupled with high-speed digital-to-analog and analog-to-digital converters to achieve Tbit/s speeds. As coherent technology matures, it will eventually become feasible for short-reach transmission. In this context, self-coherent systems have emerged as an intermediary solution, offering advantages over traditional IM/DD systems. While comprehensive review studies exist on self-coherent transceivers, they do not cover recent advances in phase retrieval methods for short-reach optical communications. This review article highlights recent developments in cost-effective self-coherent detection for short-reach systems through comparing the benefits of single sideband (SSB) transmission and Kramers-Kronig detection to carrier-assisted phase retrieval, the Gerchberg-Saxton (GS) algorithm, and the transport of intensity equation (TIE) method.