Machine learning (ML) sees an increasing prevalence of being used in the internet-of-things (IoT)-based smart grid. However, the trustworthiness of ML is a severe issue that must be addressed to accommodate the trend of ML-based smart grid applications (MLsgAPPs). The adversarial distortion injected into the power signal will greatly affect the system's normal control and operation. Therefore, it is imperative to conduct vulnerability assessment for MLsgAPPs applied in the context of safety-critical power systems. In this paper, we provide a comprehensive review of the recent progress in designing attack and defense methods for MLsgAPPs. Unlike the traditional survey about ML security, this is the first review work about the security of MLsgAPPs that focuses on the characteristics of power systems. We first highlight the specifics for constructing the adversarial attacks on MLsgAPPs. Then, the vulnerability of MLsgAPP is analyzed from both the aspects of the power system and ML model. Afterward, a comprehensive survey is conducted to review and compare existing studies about the adversarial attacks on MLsgAPPs in scenarios of generation, transmission, distribution, and consumption, and the countermeasures are reviewed according to the attacks that they defend against. Finally, the future research directions are discussed on the attacker's and defender's side, respectively. We also analyze the potential vulnerability of large language model-based (e.g., ChatGPT) power system applications. Overall, we encourage more researchers to contribute to investigating the adversarial issues of MLsgAPPs.