Below are links to official playlists that The Weather Channel released. I obtained most of them from The Weather Channel and BMI (the publisher they license music from). The rest were sent to me by other viewers. If you have one to add, please contact me.
A few years ago, long-time TWCC contributor Layne B. sent me several lists of music that he put together (covering December 1987 to Christmas 1991), based on what he remembers hearing in his area. While not "official," they are often more complete and accurate than what I have below during that time.