I didn't have to think long and hard about which song to start off with: River of Sorrow sprang to mind almost immediately. I love Antony Heggarty's lyricism, voice, and delivery; and this song's a very good example of his talent.
1) Antony and the Johnsons - River of Sorrow
For song choice No. 2, I particularly like the chaos and cacophony into which this song seemingly descends towards the end. And look, Conor Oberst even finishes on a classic guitar smashup!
2) Bright Eyes - Road to Joy
The original version of my third offering is 8 minutes long and doesn't feature Sinéad O'Connor, but I couldn't find a good quality video for it so here's the excised version:
3) Afro Celt Sound System ft. Sinéad O'Connor - Release
My fourth choice is included in this list mainly because I like the simplicity of the video, while the song itself I think may be throwaway, yet still catchy.
4) Ladytron - Runaway
I was going to try and do this without taking recourse to any of my all-time favourite artists' back catalogues, but I've found I got rather stuck on the letter R quite quickly. So, my last favourite song starting with R is
5) David Sylvian - Ride