Understanding various humour styles is essential for comprehending the multifaceted nature of humour and its impact on fields such as psychology and artificial intelligence. This understanding has revealed that humour, depending on the style employed, can either have therapeutic or detrimental effects on an individual's health and relationships. Although studies dedicated exclusively to computational-based humour style analysis remain somewhat rare, an expansive body of research thrives within related task, particularly binary humour and sarcasm recognition. In this systematic literature review (SLR), we survey the landscape of computational techniques applied to these related tasks and also uncover their fundamental relevance to humour style analysis. Through this study, we unveil common approaches, illuminate various datasets and evaluation metrics, and effectively navigate the complex terrain of humour research. Our efforts determine potential research gaps and outlined promising directions. Furthermore, the SLR identifies a range of features and computational models that can seamlessly transition from related tasks like binary humour and sarcasm detection to invigorate humour style classification. These features encompass incongruity, sentiment and polarity analysis, ambiguity detection, acoustic nuances, visual cues, contextual insights, and more. The computational models that emerge contain traditional machine learning paradigms, neural network architectures, transformer-based models, and specialised models attuned to the nuances of humour. Finally, the SLR provides access to existing datasets related to humour and sarcasm, facilitating the work of future researchers.