In this letter, we present a study of the confinement properties of point-defect resonators in finite-size photonic-bandgap structures composed of aperiodic arrangements of dielectric rods, with special emphasis on their use for the design of cavities for particle accelerators. Specifically, for representative geometries, we study the properties of the fundamental mode (as a function of the filling fraction, structure size, and losses) via two-dimensional and three-dimensional full-wave numerical simulations, as well as microwave measurements at room temperature. Results indicate that for reduced-size structures, aperiodic geometries exhibit superior confinement properties by comparison with periodic ones.