Holomorphic functions play a fundamental role in operator theory, particularly through their Cauchy formula, which is essential for defining the Riesz-Dunford functional calculus. Extensions of holomorphic functions of a complex variable include slice hyperholomorphic functions and axially Fueter regular functions. These two function classes naturally emerge from the Fueter extension theorem, which provides a two-step procedure for extending holomorphic functions to axially Fueter regular functions. In the first step, by applying the so-called slice operator, holomorphic functions of one complex variable are extended to slice hyperholomorphic functions. The corresponding Cauchy formula for these functions leads to the definition of the S-functional calculus for bounded operators with commuting components. The second step of the procedure involves applying the Laplace operator in four real variables to slice hyperholomorphic functions, resulting in axially Fueter regular functions. The Cauchy formula associated with this class of functions leads to the monogenic functional calculus, which was developed by McIntosh and collaborators. In this survey, we concentrate on the second step of the Fueter construction. The various function spaces and the corresponding functional calculi that emerge from the factorization of the Laplace operator in four real variables give rise to the concept of fine structures in spectral theories on the S-spectrum. These fine structures provide a unified framework that connects different function spaces, allowing for a deeper understanding of their interrelations and their applications in operator theory.