Quantized spin-wave eigenmodes in single, 16 nm thick and 0.75 to 4 mum wide square permalloy islands with a fourfold closure domain structure have been investigated by microfocus Brillouin light scattering spectroscopy and time resolved scanning magneto-optical Kerr microscopy. Up to six eigenmodes were detected and classified. The main direction of the spin-wave quantization in the domains was found to be perpendicular to the local static magnetization. An additional less pronounced quantization along the direction parallel to the static magnetization was also observed.