The propagation of coherent spin waves in permalloy microstrips was investigated using a heterodyne-magneto-optical Kerr effect technique. The two-dimensional imaging reveals the mode interference with a high-order spin-wave mode along the microstrip width. The spin-wave dispersion relations across the microstrip width were deduced from fast Fourier transformations on the complex spin-wave amplitude of the one-dimensional scan, and multiple spin-wave modes were observed at the edges of the microstrips. These experimental results provide direct evidence for the quantization of spin-wave dispersion due to the finite width of the microstrips and enables us to evaluate the attenuation length for each quantized spin-wave mode.