As the automotive and telecommunication industries advance, more vehicles are becoming connected, leading to the realization of intelligent transportation systems (ITS). Vehicular ad-hoc network (VANET) supports various ITS services, including safety, convenience, and infotainment services for drivers and passengers. Generally, such services are realized through data sharing among vehicles and nearby infrastructures or vehicles over multi-hop data routing mechanisms. Vehicular data routing faces many challenges caused by vehicle dynamicity, intermittent connectivity, and diverse application requirements. Consequently, the software-defined networking (SDN) paradigm offers unique features such as programmability and flexibility to enhance vehicular network performance and management and meet the quality of services (QoS) requirements of various VANET services. Recently, VANET routing protocols have been improved using the multilevel knowledge and an up-to-date global view of traffic conditions offered by SDN technology. The primary objective of this study is to furnish comprehensive information regarding the current SDN-based VANET routing protocols, encompassing intricate details of their underlying mechanisms, forwarding algorithms, and architectural considerations. Each protocol will be thoroughly examined individually, elucidating its strengths, weaknesses, and proposed enhancements. Also, the software-defined vehicular network (SDVN) architectures are presented according to their operation modes and controlling degree. Then, the potential of SDN-based VANET is explored from the aspect of routing and the design requirements of routing protocols in SDVNs. SDVN routing algorithms are uniquely classified according to various criteria. In addition, a complete comparative analysis will be achieved to analyze the protocols regarding performance, optimization, and simulation results. Finally, the challenges and upcoming research directions for developing such protocols are widely stated here. By presenting such insights, this paper provides a comprehensive overview and inspires researchers to enhance existing protocols and explore novel solutions, thereby paving the way for innovation in this field.