The analysis of partial differential equations (PDEs) is a highly adaptable field within mathematics. It is appealing from an abstract perspective due to the depth and beauty of its fundamental questions, and it has practical applications in various disciplines such as physics, biology, and economics. For instance, PDEs are used in efforts to calculate the “age of the earth,” in studies on aerodynamics, and in models of opinion formation. Regularity theory for PDEs, one of the most refined and intricate research areas related to these equations, briefly examines the structure of a given equation. From its intrinsic properties, it uncovers universal characteristics of the solutions. For example, the ellipticity of an equation ensures that the derivative of its solutions exists and is (slightly more than) continuous. Our research is particularly interested in the implications of the failure or absence of these intrinsic properties. In other words, we aim to understand what can be said about the solutions of an equation when very basic and universally accepted assumptions are removed. We explore questions such as how the graphs of the solutions behave, whether they admit cusps or have corners, and if the ‘acceleration’ of these graphs can arbitrarily increase.