2013-07-31 (updated 2016-08-30)
WinG was supported for:
Windows 3.1
Windows 95
Windows 98
Windows NT 4.0

It was not included in Windows 98 SE nor Windows 2000 nor any later Windows OS in favor of DirectX (6) which was shipped and other integrated Win32 API functionality.
However, later operating systems will usually run WinG games by adding wing.dll, wing32.dll, wingde.dll, wingdib.drv and wngpal.wnd to the system32 directory. Many WinG games install these files during installation.

When WinG first runs, a screen of red curved lines is drawn and starts looking wavy and glitchy. This is an automated test that WinG uses to determine the capabilities of the graphics hardware. The results are saved and 'never need to be run again'. This was in effect, an early version of DxDiag.
UKNOWNS
What happens if a person changes their hardware?
Can the WinG test be forced or run manually?

Chris Hecker assisted by Todd Laney created the WinG API. It was apparently not some concerted effort sanctioned by Microsoft to improve Windows gaming. Rather, Hecker was hacking a variety of broken graphics drivers and getting some games to run 10 to 100 times faster than existing performance under Windows. Anyways, Management found out and slapped a deadline on the daring enterprise to make a deployable product by 1995-12-25. Activision, ID, Origin, Mindscape, and Disney were wooed into making Windows games based on the promise of great gaming performance under Windows95, even though WinG was not yet ready. Disney was betting the most, 1 million copies of the Lion King Game would be pre-installed Compaq Presario Computers ready to be shipped and wrapped as gifts for children that Christmas. Compaq intended and advertised this computers for families and especially kids. But, Compaq had changed the Presario specs just before the shipping date. They had a new Cirrus video chip and driver, not yet patched by the WinG team. In fact, they'd never before heard of the new Cirrus video chip. All 1 million copies of The Lion King on Presario, 1 million Christmas presents, were bluescreening. An army of Disney suits descended on Microsoft. Certain WinG team members actually accepted some responsibility for the situation, but in an unusual occurrence, upper and middle management took the heat. The WinG team was actually congratulated for a mostly successful effort because, despite the Lion King disaster, many other companies had been persuaded to make Windows95 games. The team was however, ask to help ensure something like this never happened again. Weekly meetings between game dev's, hardware dev, and WinG devs as the WinG API was being developed. "WinG" was actually very short lived, this new development process lead to Microsoft's DirectX. Ironically, the first WinG game ever developed was DOOM, but it was never shipped as a product.

WinG was officially shipped 1994-09-21

2013-07-31
Can the WinG test be forced or run manually?

Likely by deleting or modifying some .ini file system32. Win.ini (predecessor of the registry) was common place to store lots of random data as was system.ini, so those are likely candidates where the configuration would be stored if there's no WinG specific ini file.