Transfer learning has proven to be crucial in advancing the state of speech and natural language processing research in recent years. In speech, a model pre-trained by self-supervised learning transfers remarkably well on multiple tasks. However, the lack of a consistent evaluation methodology is limiting towards a holistic understanding of the efficacy of such models. SUPERB was a step towards introducing a common benchmark to evaluate pre-trained models across various speech tasks. In this paper, we introduce SUPERB-SG, a new benchmark focusing on evaluating the semantic and generative capabilities of pre-trained models by increasing task diversity and difficulty over SUPERB. We use a lightweight methodology to test the robustness of representations learned by pre-trained models under shifts in data domain and quality across different types of tasks. It entails freezing pre-trained model parameters, only using simple task-specific trainable heads. The goal is to be inclusive of all researchers, and encourage efficient use of computational resources. We also show that the task diversity of SUPERB-SG coupled with limited task supervision is an effective recipe for evaluating the generalizability of model representation.