I read fics of my favorite ships because I just can't get enough of them on the actual shows. However, I seem to ship unpopular pairings because I'm attracted to the unconventional, broken relationships.
Examples of unconventional ships:
Josh/Amy (The West Wing)
Lorelai/Christopher (Gilmore Girls)
Buffy/Angel and Buffy/Spike -- I know these are popular ships, but I still think both relationships are so messed up.