Speech-centric machine learning systems have revolutionized many leading domains ranging from transportation and healthcare to education and defense, profoundly changing how people live, work, and interact with each other. However, recent studies have demonstrated that many speech-centric ML systems may need to be considered more trustworthy for broader deployment. Specifically, concerns over privacy breaches, discriminating performance, and vulnerability to adversarial attacks have all been discovered in ML research fields. In order to address the above challenges and risks, a significant number of efforts have been made to ensure these ML systems are trustworthy, especially private, safe, and fair. In this paper, we conduct the first comprehensive survey on speech-centric trustworthy ML topics related to privacy, safety, and fairness. In addition to serving as a summary report for the research community, we point out several promising future research directions to inspire the researchers who wish to explore further in this area.