At first there were 2D graphics, and depth was no use.
Then Z-axis popped out and suddenly people wondered how to solve the visibility problem. The region of memory where you store object's depth information is the Z-buffer.
This information can be more or less precise, and this is why there are 16, 24 and 32 bit buffers (and actually many other variants, but that's another story)
Today the standard is 24+8 stencil buffer. And this is why the later was dumped in early 2000s and the former is now in an agonizing situation.
Ok, after this creepy introduction I can start talking of my findings, that mainly focus on ATI cards, but aren't all that different even for nvidia ones.
Everything started to fall apart with driver 7.5, which first broke dithering used by 16 bit games to counter the lower precision.
Later another blow seems to have come with 7.12.
An excerpt from one of the AMD forums moderator claims it was even due to lack of support in the newer hardware itself.
Unfortunately the original thread (whose title can be seen here) is no longer available for viewing, along with many many other useful references of those times that have been purged some months ago from their updated boards.
Google still seems to keep some sort of memory, but I can't access it.
In the meantime, I found some really accurate (and complementary) explanations of the broken functionalities.
Another allegedly malfunctioning one should be 8 bit Palletized Texture support.
Fast forward to our times ultimately, 16 bit Zbuffer is still nonworking.
A lot of fixes around the net suggest to use a different atiumdag.dll, which I did unsuccessfully until catalyst 9.8 one, the older that doesn't throw an error on my 7x64 SP1 with HD5770.
Then others assert that is has even been already fixed..
But tl;dr I could only get 2 things to work with Omikron (aside of its specific injector): DXWnd Clean ZBUFFER @0.0 fix and dgvoodoo.
Unless somebody come out with something that merge the previously underlined stuff (the main course of this post tbh) these are the only general solutions.
EDIT: tested on a GTX 680 and.. well, even if sometimes it wasn't running at 144 steady FPS in all its power, it had no glitches at least. Kudos to them.
EDIT2: this might help? EDIT3: see probably here