Recent work on the growth of thin metal films on quasicrystalline substrates has indicated the formation of so-called “magic height” islands with multiples of 4 atomic layers (AL) arising as a result of quantum size effects, which lead to enhanced stability. Here the results of a study are reported of Bi deposition on i-Al-Pd-Mn using medium-energy ion scattering to characterize the island thickness and the structural arrangement of Bi atoms within the islands. In addition, data were taken from annealed surfaces after Bi cluster desorption to leave a single aperiodic monolayer of Bi at the surface. Scattered-ion energy spectra from the Bi islands are consistent with a single Bi monolayer covered with mainly 4 AL islands for both 1.8 and 3.2 monolayer equivalent coverages but with some occupation of 2 and 8 Al islands as well. The angular dependence of the scattered-ion intensity (“blocking curve”) from Bi has been compared with simulations for various models of both rhombohedral Bi and a distorted “black-phosphorus”-like structure. The data demonstrate bilayer formation within the Bi islands. In the case of the aperiodic Bi monolayer, the blocking curves from substrate scattering are found to be inconsistent with two high-symmetry sites on the quasicrystalline surface that theory indicates are energetically favorable but do not exclude the formation of pentagonal arrangements of Bi atoms as seen in other recent experimental work.