This list concentrates on music videos that were significant in some way in themselves - it is not a list of the most significant music (though some of the songs undoubtedly were very influential).
"Bohemian Rhapsody" by Queen (early music video, long, progressive and bizarre, making it unusual for single. Bohemian Rhapsody was actually the first music 'video'.)
"Every Breath You Take" by The Police (notable as the most famous video to use black and white footage as a stylistic choice)
"Girls on Film" by Duran Duran (early example of a controversial video clip for its nudity and depiction of sexual fetishes.)
"Union of the Snake" by Duran Duran (the first song released to MTV before its radio release -- caused a huge ruckus at the time)
"When Doves Cry" by Prince (extremely popular and controversial for sexual content)
"Thriller" by Michael Jackson (unusually cinematic for its time, including use of at that time 'cutting edge' special effects. This video was also unusually long (15 minutes), and framed the actual musical content within a wider narrative context)
"What I Got" by Sublime (breakout success occurring only a few weeks after the death of singer Bradley Nowell, one of the longest-lasting videos on various countdowns in history)
"Star Guitar" by Chemical Brothers (Used a huge amount of digital editing to create a squence of landscape passing by as viewed from a train that synchronized with the beat of the music)