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In this paper, a nonlocal homogenization model is proposed for the analysis of the spectrum of natural modes on sub-wavelength mushroom-type high-impedance surfaces composed of a capacitive grid connected to a grounded wire-medium (WM) slab. Modal characteristics of mushroom structures are studied in conjunction with the surface-wave and leaky-wave propagation on WM slabs based on local and nonlocal homogenization models, showing the importance of spatial dispersion (SD) in WM. It is shown that mushroom structures support proper real (bound) forward and backward modes, whose dispersion determines the stopband properties of the mushroom structure, and proper (exponentially decaying from the surface) and improper (exponentially growing from the surface) complex leaky-wave modes related to the backward and forward radiation, respectively. Results obtained by different homogenization models are compared leading to important conclusions. Specifically, an interesting observation concerns the mushroom structures with short vias, wherein the SD of the WM slab is significantly reduced, and the results of local and nonlocal homogenization models are in excellent agreement.