Side-channel attacks on microprocessors, like the RISC-V, exhibit security vulnerabilities that lead to several design challenges. Hence, it is imperative to study and analyze these security vulnerabilities comprehensively. In this paper, we present a brief yet comprehensive study of the security vulnerabilities in modern microprocessors with respect to side-channel attacks and their respective mitigation techniques. The focus of this paper is to analyze the hardware-exploitable side-channel attack using power consumption and software-exploitable side-channel attacks to manipulate cache. Towards this, we perform an in-depth analysis of the applicability and practical implications of cache attacks on RISC-V microprocessors and their associated challenges. Finally, based on the comparative study and our analysis, we highlight some key research directions to develop robust RISC-V microprocessors that are resilient to side-channel attacks.