Smart Grid (SG) is a multidisciplinary concept related to the power system update and improvement. SG implies real-time information with specific communication requirements. System reliability relies on the best capabilities for monitoring and controlling the grid. Among other aspects, SG applications involve three main challenges, sufficient real-time capable measurement units, managing large data sets, and two-way low-latency communications. Considering fault detection and classification a key factor to SG reliability, this work provides a systematic review of SG faults from the most significant research databases and state-of-the-art research papers aiming at creating a comprehensive classification framework on the relevant requirements. This paper includes in detail the classification of different fault scenarios in a comprehensive framework that involves system-level of application, e.g., transmission, distribution, commercial, DG, and EV. To this end, We analyze and indicate relevant topics for future developments related to the monitoring and fault detection and classification in SG systems.