In the order that I remember
  1. Love Cats by The Cure
  2. Lioness by Song: Ohia
  3. Year of the Tiger by St. Vincent
  4. Wildcat by Ratatat
  5. Batcat by Mogwai
  6. Leaf House by Animal Collective
  7. Little Lion Man by Mumford & Sons
  8. Cat People by David Bowie
  9. Animal Language by Lou Reed
  10. Year of the Cat by Psapp
  11. Black Cat by Janet Jackson
  12. Yesterdays by Billie Holiday
    Because this was featured in Fritz the Cat
  13. Everybody Wants To Be a Cat from The Aristocats
  14. Tommy the Cat by Primus
  15. Stray Cat Strut by The Stray Cats
    Because this is a given
  16. Skeleton Key by Margot and the Nuclear So And So's
  17. Meow Meow Lullaby by Nada Surf
  18. Smelly Cat by Phoebe Buffay
    Suggested by   @Melodie
  19. Eye of the Tiger
    Suggested by   @MichaelRose
  20. What's new, pussycat? By Tom Jones
    Suggested by   @holly70
  21. The Meow Mix commercial jingle
    Suggested by   @e
  22. Smelly Cat - Phoebe Buffay
    Suggested by   @shreyasaurus