Humour, as a complex and often esoteric language form, is derived from myriad aspects of life, whilst existing work on computational humour has focussed almost exclusively on short, self-contained pun-based jokes. In this work, we investigate whether the ability of Large Language Models (LLMs) to explain humour is dependent on the particular humour form. Specifically, we aim to compare models on simple pun-based jokes and more complex contemporary topical humour that requires knowledge of real-world entities and events. In doing so, we curate a dataset of 600 jokes, evenly split across 4 joke types, and manually write high-quality textual explanations, including heterographic and homographic puns, as well as contemporary internet humour and topical jokes where adequate understanding is reliant on reasoning beyond "common sense", rooted instead in world knowledge regarding news events, politics, pop culture, and more. Using this dataset, we compare the zero-shot abilities of a range of open- and closed-source 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 are capable of reliably generating adequate explanations of all joke forms (inc. reasoning models), further highlighting the narrow focus of most works in computational humour processing on overly simple humour forms.