Various polygons intersect and overlap where they shouldn't. This is exceedingly apparent with the two shown characters' outfits. It can also be seen in the corners and edges in the architecture in the background of the image.
If I run the game in DX11 mode, this is not a problem; however, alt-tabbing and trying to run the game in fullscreen mode is just a pain in the neck as it seems the screen resolution settings in STO's DX11 implementation are totally borked.
Enabling Anti-Aliasing reduces the severity of this issue, but it can sometimes still be apparent, particularly with neighboring polygons of highly contrasting colors. My graphics card also can't support a playable framerate with AA enabled, so I can't work around the issue this way.
The normal answer I would see to this kind of problem is "Update your display driver." I am playing on a laptop with an AMD Mobility Radeon HD 3450 with Catalyst software version 12.6 (the final supported version downloaded from AMD as of the time of this writing). There is no computer manufacturer-supplied display driver for my system compatible with Windows 7.
The only theories I have are that something is wrong with z-buffer resolution on my system. I don't know how to fix it. I know it must be possible to correct as there was a time several years ago when this was not a problem, and none of the hardware in my system has changed.