In some ways interchangeable terms, in some ways three different things.
music library - generally a term for a company (although may be contained within a company as well) that has a collection of music to be "rented" or licensed out for third party use.
synch catalogue - a catalogue that may be in a music library, record company, publishing house or even privately owned that is available for sound to picture use. Synch refers to the use of music when "synchronised" to the moving image. Slightly different set of rights and permissions generally go along with a synch.
song catalogue - more obvious. Songs! Not instrumentals or soundtrack pieces (although they CAN be). Generally artist related compositions or music in song format.