Great idea for a post. I have chased songs for months or years myself. Today, with the internet and the various music services, It is much easier to find things. A couple of mine: When I was in H.S. in 1961, I heard a brassy, uptempo version of Pomp and Circumstance on the radio. It was by "Adrian Kimberly". I had to order it from the nearest record store, because they didn't carry it. I still have the 45 and only recently learned Kimberly was actually Don Everly.
In the 1990's I heard a song as background on a PBS station. I was obsessed with it and called the station to get the name. They were no help. It was a very sweeping, emotional tune and it then showed up as the music constantly used as the theme for the TV broadcast of the Olympic Games. It was also used as background music for several movie trailers. I finally was able to determine that it was a song called "The Dragon's Heart" from the movie Dragon: The Bruce Lee Story. Soundtrack by Randy Edelman.