The only two non-routine maintinance related problem that I had with the car were steering rack and CV joints.
The power steering rack fix was done ourselves, using a junkyard rack. the repair was a success.
The CV joints were wearing to the point where they would snap crackle and pop even on slight turns. Then we sold the car. By that time, the seats were looking rough, the light blue paint was peeling and it had been hit in the rear.
As a young driver, I drove that car all over the place. I thought it was a faster and roomier car than the Escorts and Chevettes that I'd been in. Looking back, I should've just fixed the cv joints and kept on driving it. Other Omni owners seem to have lots of complaints and hard-luck stories, but I thought the car was a pretty good idea. They've all, but disappeared from the roads around here, but I'd seriously consider buying another if one came my way.
I have the same car and I agree, the CV joints were horrible. I did get them replaced and she ran like an angel... until I rear-ended a dump-truck. She got totaled. I'm currently about to buy another Omni in a later model with more features, but for the same price. I just loved that little car.
I have an '83 Omni, and mine makes a pretty disconcerting clicking noise on right turns, my dad/mechanic said it was the CV joints. If rust wasn't eating away at the frame on mine, I'd get them replaced and drive it for as long as is humanly possible. I love my Omni.