Software-Defined Networking (SDN) is a modern paradigm that provides a platform for implementing reliable, centrally managed, and automated security solutions for conventional and new generation networks, such as IoT, cloud computing, 5G/6G mobile communication networks, and vehicular communications. In these complex systems, manual security operations can delay or obstruct the identification, mitigation, and prevention of ever-increasing sophisticated threats. Thus, the idea of security automation for networks using the SDN paradigm has become fundamental, given that SDN was created to facilitate the operation and management of complex networks with minimal human intervention, which is considered error-prone. This survey studies the state-of-the-art research efforts concerned with security automation in SDN environments. We identified and ranked various classes of security solutions with different levels of automation and complexity. The level of automation is measured using four well-defined qualitative parameters: self-healing, self-adaptation, self-configuration, and self-optimization. The complexity is characterized by the amount of processing and storage resources and implementation requirements. This work represents the first endeavor to analyze the level of automation and complexity of security solutions in SDN environments. Our findings reveal important advances in the area of security automation in SDN. However, there are still several open problems and challenges, which we detail in this work.