Storage services in data centers continuously make decisions, such as for cache admission, prefetching, and block allocation. These decisions are typically driven by heuristics based on statistical workload properties like temporal locality or common file sizes. The quality of decisions can be improved through application-level information such as the database operation a request belongs to. While such features can be exploited through application hints (e.g., explicit prefetches), this process requires manual work and is thus only viable for the most tuned workloads. In this work, we show how to leverage application-level information automatically, by building on distributed traces that are already available in warehouse-scale computers. As these traces are used for diagnostics and accounting, they contain information about requests, including those to storage services. However, this information is mostly unstructured (e.g., arbitrary text) and thus difficult to use. We demonstrate how to do so automatically using machine learning, by applying ideas from natural language processing. We show that different storage-related decisions can be learned from distributed traces, using models ranging from simple clustering techniques to neural networks. Instead of designing different models for different storage-related tasks, we show that the same model can be used as a building block for different tasks. Our models improve prediction accuracy by 11-39% over non-ML baselines, which translates to significantly improving the hit rate of a caching task, as well as improvements to an SSD/HDD tiering task, on production data center storage traces.