Massive deployment of distributed energy resources (DERs) along with innovations in information and communication technologies have changed the power system from a hierarchical structure to a more deregulated model by introducing new generations at lower levels. This change raises operational and market challenges. Local energy trading provides opportunities to manage these DERs by encouraging localised trading. The concept of local energy trading at the distribution level is widely broadcasted for implementation in the power system. In the design of electricity markets for local energy trading, a clear definition of market participants, their objective, and purpose of market clearing should be established. This design depends on the changing needs of the power system and can be performed from different viewpoints. Classification and organisation of the literature on potential designs for local energy trading can help researchers to develop their future steps properly. This study presents a comprehensive review on this topic and provides a systematic classification of the market players, market clearing objectives, and approaches. Several research works are analysed against different criteria, including scalability, overheads requirements, and network constraints management.