Update for 2020: Sticking to the original question of 16×16 versus 32×32 icons: the current recommendation should be to provide a 32×32 icon, skipping 16×16 entirely. All current browsers and devices support 32×32 icons. The icon will routinely be upscaled to as much as 192×192 depending on the environment (assuming there are no larger sizes available or the system didn’t recognize them). Upscaling from ultra low resolution has a noticeable effect so better stick to 32×32 as the smallest baseline.
For IE, Microsoft recommends 16×16, 32×32 and 48×48 packed in the favicon.ico file.
For iOS, Apple recommends specific file names and resolutions, at most 180×180 for latest devices running iOS 8.
Android Chrome primarily uses a manifest and also relies on the Apple touch icon.
IE 10 on Windows 8.0 requires PNG pictures and a background color and IE 11 on Windows 8.1 and 10 accepts several PNG pictures declared in a dedicated XML file called browserconfig.xml.
Safari for Mac OS X El Capitan introduces an SVG icon for pinned tabs.
Some other platforms look for PNG files with various resolutions, like the 96×96 picture for Google TV or the 228×228 picture for Opera Coast.
Look at this favicon pictures list for a complete reference.
TLDR: This favicon generator can generate all these files at once. The generator can also be implemented as a WordPress plugin. Full disclosure: I am the author of this site.