Twin-width is a recently introduced graph parameter based on the repeated contraction of near-twins. It has shown remarkable utility in algorithmic and structural graph theory, as well as in finite model theory -- particularly since first-order model checking is fixed-parameter tractable when a witness certifying small twin-width is provided. However, the behavior of twin-width in specific graph classes, particularly cubic graphs, remains poorly understood. While cubic graphs are known to have unbounded twin-width, no explicit cubic graph of twin-width greater than 4 is known. This paper explores this phenomenon in regular and near-regular graph classes. We show that extremal graphs of bounded degree and high twin-width are asymmetric, partly explaining their elusiveness. Additionally, we establish bounds for circulant and d-degenerate graphs, and examine strongly regular graphs, which exhibit similar behavior to cubic graphs. Our results include determining the twin-width of Johnson graphs over 2-sets, and cyclic Latin square graphs.