We propose a method for learning the structure of variable-order CRFs, a more flexible variant of higher-order linear-chain CRFs. Variable-order CRFs achieve faster inference by including features for only some of the tag n-grams. Our learning method discovers the useful higher-order features at the same time as it trains their weights, by maximizing an objective that combines log-likelihood with a structured-sparsity regularizer. An active-set outer loop allows the feature set to grow as far as needed. On part-of-speech tagging in 5 randomly chosen languages from the Universal Dependencies dataset, our method of shrinking the model achieved a 2-6x speedup over a baseline, with no significant drop in accuracy.