Secure keyword search is a prevailing search service offered in outsourced environments. However, with the increasingly severe security vulnerabilities of conventional centralized outsourcing, the architecture of secure keyword search, with searchable encryption (SE) as the underlying technique, has recently shifted from cloud-centered models to blockchain-assisted models. Existing surveys commonly fail to capture such an evolution and the corresponding benefits. What on earth does blockchain bring about and what are the unexplored challenges? This survey provides a systematic review of secure keyword search over outsourced data from cloud to blockchain-assisted architectures. We propose a taxonomy assorting present studies, depending on whether cloud/blockchain and data sharing are included, in which blockchain-assisted architecture is further divided into blockchain-side and cloud-side keyword search, respectively. Technically, we conclude five types of representative SE techniques with fitting architectures, either cryptographic-based or hardware-dependent. Notably, we propose comprehensive methodologies to select relevant papers, discuss, and compare existing schemes regarding functionalities, security, efficiency, and fairness (up to 21 compared items). Finally, open issues and potential research directions are identified for future work. We aspire to help pave the way for addressing the theoretical and empirical aspects of secure keyword search and full-fledged real-world implementation of blockchain-based keyword search applications.