Podcasts have become daily companions for half a billion users. Given the enormous amount of podcast content available, highlights provide a valuable signal that helps viewers get the gist of an episode and decide if they want to invest in listening to it in its entirety. However, identifying highlights automatically is challenging due to the unstructured and long-form nature of the content. We introduce Rhapsody, a dataset of 13K podcast episodes paired with segment-level highlight scores derived from YouTube's 'most replayed' feature. We frame the podcast highlight detection as a segment-level binary classification task. We explore various baseline approaches, including zero-shot prompting of language models and lightweight finetuned language models using segment-level classification heads. Our experimental results indicate that even state-of-the-art language models like GPT-4o and Gemini struggle with this task, while models finetuned with in-domain data significantly outperform their zero-shot performance. The finetuned model benefits from leveraging both speech signal features and transcripts. These findings highlight the challenges for fine-grained information access in long-form spoken media.