I've seen horizontal line artifacts occur in some high scan modes when the custom chips get a tad warm.
Is your system in a normal case or in some sort of tower?
Extra cooling always helps, but it can be tricky to get a well defined airflow in a desktop A1200 type case.
What sort of accelerator are you using (I can't see if you have written this in your sig or something as I write this so apologies if it is written there) ?
-sdit-
Anyway, if there is room in your case, you can mount some small cooling ribs on the custom chips (assuming you aren't using some expansion that clips on them.
An 030 card typically shouldn't generate that much heat (some do, though).
3.5 inch hard drives contribute a lot of heat, but you say yours is external, so I guess it isn't that.
A long time ago, I ran my BlizzPPC in a desktop case. To address the cooling (and to give somewhere to put my external CD) I made a small 'dock' for the 1200 to sit on, that was exactly as tall as the CD and as large as the 1200 base.
The CD sat in the left hand side, tray front facing and the large heatsink I installed on the 040 poked into the remaining space on the right hand side.
I made sure the overall dock had no unwanted gaps or holes and then installed several large fans as air intakes at the back (with fine mesh covered holes to keep the dust out). The overall effect was to pull cold air in at the back. The only place for this air to exhaust was through the 1200, via the trapdoor and the underside vents at the front and ultimately out through the vents at the back.
Overall it worked quite well. Later I got a tower, so the cooling was no longer an issue.
