The Barenaked Ladies play the song Pinch Me from their album Maroon during the episode but I don't know if this is the song you are asking about.
In season 5 episode 4, the episode is called Siren Song
My love is like a red, red rose
Alice Blue Gown
The Piper - ABBA song - was created in 1980.
The XX-Shelter
That song is Chances by the band Athlete.
The song was "One Step Closer". It was featured in the ER episode "On the Beach."
The 'Arnold Rimmer Song' is from the season 7 episode 'Blue'.
Crispen St. Peter did a song in the 60's called "The Pied Piper"
The song at the end of 'The Christmas Invasion' in Series 2, Episode 1 is "Song For Ten" from the Doctor Who: Original Television Soundtrack.
low - point of disgust
Iron- Woodkid