A well-functioning smart grid is an essential part of an efficient and uninterrupted power supply for the key enablers of smart cities. To effectively manage the operations of a smart grid, there is an essential requirement for a seamless wireless communication system that provides high data rates, reliability, flexibility, massive connectivity, low latency, security, and adaptability to changing needs. A contemporary review of the utilization of emerging 6G wireless communication for the major applications of smart grids, especially in terms of massive connectivity and monitoring, secured communication for operation and resource management, and time-critical operations, are presented in this paper. This article starts with the key enablers of the smart city, along with the necessity of the smart grid for the key enablers of it. The fundamentals of the smart city, smart grid, and 6G wireless communication are also introduced in this paper. Moreover, the motivations to integrate 6G wireless communication with the smart grid system are expressed in this article as well. The relevant literature overview, along with the novelty of this paper, is depicted to bridge the gap of the current research works. We describe the novel technologies of 6G wireless communication to effectively perform the considered smart grid applications. Novel technologies of 6G wireless communication have significantly improved the key performance indicators compared to the prior generation of the wireless communication system. A significant part of this article is the contemporary survey of the considered major applications of a smart grid that is served by 6G. In addition, the anticipated challenges and interesting future research pathways are also discussed explicitly in this article. This article serves as a valuable resource for understanding the potential of 6G wireless communication in advancing smart grid applications and addressing emerging challenges.