Support vector machines (SVMs) are well-known machine learning algorithms for classification and regression applications. In the healthcare domain, they have been used for a variety of tasks including diagnosis, prognosis, and prediction of disease outcomes. This review is an extensive survey on the current state-of-the-art of SVMs developed and applied in the medical field over the years. Many variants of SVM-based approaches have been developed to enhance their generalisation capabilities. We illustrate the most interesting SVM-based models that have been developed and applied in healthcare to improve performance metrics on benchmark datasets, including hybrid classification methods that combine, for instance, optimization algorithms with SVMs. We even report interesting results found in medical applications related to real-world data. Several issues around SVMs, such as selection of hyperparameters and learning from data of questionable quality, are discussed as well. The several variants developed and introduced over the years could be useful in designing new methods to improve performance in critical fields such as healthcare, where accuracy, specificity, and other metrics are crucial. Finally, current research trends and future directions are underlined.