I was triggered to start a topic on this when I saw this YouTube video.
Top 10 Cheesiest One-Hit Wonders of the 1980s
The "10 cheesiest" part is debatable (at least on some), and despite some not being my cup of tea, they're still memorable one-hit wonders, and representative of the era.
2 of the 10 I actually thought were Hall & Oates songs, until I watched this!
I heard this song at least a year before I saw Rocky Horror:
"I Do Thee Rock", by Tim Curry(1979)
All the references to various crazy third-world dictators and celebrities... things have unfortunately changed but in many ways stayed the same in the world.