Humour, as a complex language form, is derived from myriad aspects of life, whilst existing work on computational humour has focussed almost exclusively on short pun-based jokes. In this work, we investigate whether the ability of Large Language Models (LLMs) to explain humour depends on the particular humour form. We compare models on simple puns and more complex topical humour that requires knowledge of real-world entities and events. In doing so, we curate a dataset of 600 jokes split across 4 joke types and manually write high-quality explanations. These jokes include heterographic and homographic puns, contemporary internet humour, and topical jokes, where understanding relies on reasoning beyond "common sense", rooted instead in world knowledge regarding news events and pop culture. Using this dataset, we compare the zero-shot abilities of a range of LLMs to accurately and comprehensively explain jokes of different types, identifying key research gaps in the task of humour explanation. We find that none of the tested models (inc. reasoning models) are capable of reliably generating adequate explanations of all joke types, further highlighting the narrow focus of most works in computational humour on overly simple joke forms.