With the rapid amassing of spatial-temporal (ST) ocean data, many spatial-temporal data mining (STDM) studies have been conducted to address various oceanic issues, including climate forecasting and disaster warning. Compared with typical ST data (e.g., traffic data), ST ocean data is more complicated but with unique characteristics, e.g., diverse regionality and high sparsity. These characteristics make it difficult to design and train STDM models on ST ocean data. To the best of our knowledge, a comprehensive survey of existing studies remains missing in the literature, which hinders not only computer scientists from identifying the research issues in ocean data mining but also ocean scientists to apply advanced STDM techniques. In this paper, we provide a comprehensive survey of existing STDM studies for ocean science. Concretely, we first review the widely-used ST ocean datasets and highlight their unique characteristics. Then, typical ST ocean data quality enhancement techniques are explored. Next, we classify existing STDM studies in ocean science into four types of tasks, i.e., prediction, event detection, pattern mining, and anomaly detection, and elaborate on the techniques for these tasks. Finally, promising research opportunities are discussed. This survey can help scientists from both computer science and ocean science better understand the fundamental concepts, key techniques, and open challenges of STDM for ocean science.