We have characterized the porosity of vapor-deposited amorphous solid water (ice) films deposited at 30-40 K using several complementary techniques such as quartz crystal microgravimetry, UV-visible interferometry, and infrared reflectance spectrometry in tandem with methane adsorption. The results, inferred from the gas adsorption isotherms, reveal the existence of microporosity in all vapor-deposited films condensed from both diffuse and collimated water vapor sources. Films deposited from a diffuse source show a step in the isotherms and much less adsorption at low pressures than films deposited from a collimated source with the difference increasing with film thickness. Ice films deposited from a collimated vapor source at 77 degrees incidence are mesoporous, in addition to having micropores. Remarkably, mesoporosity is retained upon warming to temperatures as high as 140 K where the ice crystallized. The binding energy distribution for methane adsorption in the micropores of ice films deposited from a collimated source peaks at approximately 0.083 eV for deposition at normal incidence and at approximately 0.077 eV for deposition at >45 degrees incidence. For microporous ice, the intensity of the infrared bands due to methane molecules on dangling OH bonds on pore surfaces increases linearly with methane uptake, up to saturation adsorption. This shows that the multilayer condensation of methane does not occur inside the micropores. Rather, filling of the core volume results from coating the pore walls with the first layer of methane, indicating pore widths below a few molecular diameters. For ice deposited at 77 degrees incidence, the increase in intensity of the dangling bond absorptions modified by methane adsorption departs from linearity at large uptakes.