EW has assembled a list of 15 shows where they consider the music to be a "costar" (as in, takes on an important role).
Beverly Hills 90210 (the original)
Buffy the Vampire Slayer
Any shows missing from this list?
One addition for me would be Scrubs, which I can only assume Zach Braff has contributed to musical selections because they frequently sound straight off of the Garden State soundtrack. Also, for the record, EW mentions "How to Save a Life" as being prominent in Grey's, but that song was actually featured first in Scrubs! I remember it distinctly.
Also, who finds it interesting that the vast majority of shows where music is viewed to be prominent is in shows geared towards young people?