The well-documented presence of texture bias in modern convolutional neural networks has led to a plethora of algorithms that promote an emphasis on shape cues, often to support generalization to new domains. Yet, common datasets, benchmarks and general model selection strategies are missing, and there is no agreed, rigorous evaluation protocol. In this paper, we investigate difficulties and limitations when training networks with reduced texture bias. In particular, we also show that proper evaluation and meaningful comparisons between methods are not trivial. We introduce BiasBed, a testbed for texture- and style-biased training, including multiple datasets and a range of existing algorithms. It comes with an extensive evaluation protocol that includes rigorous hypothesis testing to gauge the significance of the results, despite the considerable training instability of some style bias methods. Our extensive experiments, shed new light on the need for careful, statistically founded evaluation protocols for style bias (and beyond). E.g., we find that some algorithms proposed in the literature do not significantly mitigate the impact of style bias at all. With the release of BiasBed, we hope to foster a common understanding of consistent and meaningful comparisons, and consequently faster progress towards learning methods free of texture bias. Code is available at https://github.com/D1noFuzi/BiasBed.