Sounds like a hardware issue with the monitor (or more specific, the built in LCD panel; in short: time to buy a new one).
Due to the fact that the issue is isolated to a single line (even with multiple pixels in height) it's unlikely it's the actual signal (e.g. cable or connector pins).
Some issue with the connection would most likely screw up several lines and/or the whole image and it won't be isolated to exactly one line (especially with digital connections).
In a similar way, issues with the graphics card should most likely cause issues with the whole screen (like garbage/random patterns). Those shouldn't be isolated to a single line.
This could be some extreme case of stuck/dead pixels (i.e. the whole row is somehow damaged).
What you could try (this won't necessarily help because we simply don't know what caused the actual issue in the first place):
Turn off the monitor, disconnect all cables, and put it flat on your table (screen up) so it's sitting plane on the table. Let it sit there for a night, then try again.
If everything else fails, what happens if you use your finger nail and pull it over the faulty line? Don't push too hard, just so there's some pressure (typically shows some short colored trail that should disappear within 1-2 seconds). Can you see any additional patterns or "waves" that don't fit to what you see in other areas of the screen?