In this paper, we study the Hölder regularity of set-indexed stochastic processes defined in the framework of Ivanoff-Merzbach. The first key result is a Hölder-continuity Theorem derived from the approximation of the indexing collection by a nested sequence of finite subcollections. Hölder-continuity based on the increment definition for set-indexed processes is also considered. Then, the localization of these properties leads to various definitions of Hölder exponents. Moreover, a pointwise continuity exponent is defined in relation with the weak continuity property for set-indexed processes which only considers single point jumps. In the case of Gaussian processes, almost sure values are proved for the Hölder exponents. As an application, the local regularity of the set-indexed fractional Brownian motion and the Ornstein-Uhlenbeck process are proved to be constant, with probability one.