Here are the songs that I currently know:
Nananana boo boo: MP, MP, MP+HP, MK+HK, MP, MP+HP
Happy Birthday: LP x2, HK, LP, LK, LK+MK, LP x2, HK, LP, Up+HP, LK, LP x2, Up+LP, LK x3, LK+MK, HK, (Up+MP+HP) x2, Up+LP+HP, LK, Up+HP, LK
Darth Vader's Theme: HK x3, MP+HP, LK, HK, MP+HP, LK, HK...