We present a comprehensive study of sparse attention patterns in Transformer models. We first question the need for pre-training with sparse attention and present experiments showing that an efficient fine-tuning only approach yields a slightly worse but still competitive model. Then we compare the widely used local attention pattern and the less-well-studied global attention pattern, demonstrating that global patterns have several unique advantages. We also demonstrate that a flexible approach to attention, with different patterns across different layers of the model, is beneficial for some tasks. Drawing on this insight, we propose a novel Adaptive Axis Attention method, which learns—during fine-tuning—different attention patterns for each Transformer layer depending on the downstream task. Rather than choosing a fixed attention pattern, the adaptive axis attention method identifies important tokens—for each task and model layer—and focuses attention on those. It does not require pre-training to accommodate the sparse patterns and demonstrates competitive and sometimes better performance against fixed sparse attention patterns that require resource-intensive pre-training.