The first evidence of out‐of‐plane resonances in hybrid metallo‐dielectric quasi‐crystal (QC) nanostructures composed of metal‐backed aperiodically patterned low‐contrast dielectric layers is reported. Via experimental measurements and full‐wave numerical simulations, these resonant phenomena are characterized with specific reference to the Ammann‐Beenker (quasi‐ periodic, octagonal) tiling lattice geometry and the underlying physics is investigated. In particular, it is shown that, by comparison with standard periodic structures, a moderately richer spectrum of resonant modes may be excited, due to the easier achievement of phase‐matching conditions endowed by its denser Bragg spectrum. Such modes are characterized by a distinctive plasmonic or photonic behavior, discriminated by their field distribution and dependence on the metal film thickness. Moreover, the response is accurately predicted via …