We report the first evidence of out-of-plane resonances in hybrid metallo-dielectric quasi-crystal nanostructures composed of metal-backed aperiodically-patterned low-contrast dielectric layers. We numerically and experimentally characterize these resonant phenomena and investigate the underlying physics. We show that, by comparison with standard periodic structures, a richer spectrum of resonant modes may be excited. Such modes are characterized by a distinctive plasmonic or photonic behavior, discriminated by their field distribution and dependence on the metal thickness. The response is accurately predicted via computationally-affordable periodic-approximant-based numerical modeling. Finally, we explore the structure functionalization via nanosized high refractive index overlays, for resonance tuning and quality-factor (Q) enhancement, as well as its surface sensitivity to deposition of nanolayers of …