I have had varying results with different S3 Virge cards. I think it really depends on the cards maker. Number Nine cards have worked well for me but I think it was some of the STB cards that had a similar issue to what you are describing.

I have the S3 Virge DX too.
The S3 cards have a problem of elevated minimum black level.
The lines you see may be the sync pulses which might be related to the black level problem.
Some S3 Virge BIOSes fix that problem, but create others like color corruption.
It also has some picture centering problem with fundamental resolutions like 720x400.

I don't know why the S3 Virge is liked so much, I only had bad experience with it.
The Cirrus Logic 5446 performs so much better under DOS and maybe W95, although it's VBE 1.2 which is fine by me.

Try this util Jepael wrote, it fixes the black level issue on S3 cards: Re: S3 AGP Cards (and possibly others) Too Bright

S3 Virge (and in general S3 cards) are liked because the offer great compatibility under DOS -> http://gona.mactar.hu/DOS_TESTS/
Personally after wbc patched the Trio3D/2X bios I replaced the Riva 128zx card with a Speedstar A55 in my dos p1 build , same speed but more green squares according to the tests site above 😁
There is also "S3VBEFIX - PC\MS-DOS TSR fix for S3 VESA 2.0 video BIOS" wbc wrote , combo'ed with the black level fixes the S3 cards became even better!

Tried the .com files Jepael wrote, but they don't seem to work on my card.

I wonder if passing it through a Voodoo 2 PCI card would do anything to clean it up a little, I might try that, don't think it'll work.

Looking at other card options now, Maybe a MX4000 will do the job but I've got no idea what they're like in DOS