All of them on July 31, 2016
http://www.jp.playstation.com/scej/online/bcjs30018.html
http://www.jp.playstation.com/scej/online/bcjs30058.html
http://www.jp.playstation.com/scej/online/bcjs30095.html
http://www.jp.playstation.com/scej/online/pcjs50003.html
http://www.jp.playstation.com/scej/online/ucjs10107.html
http://www.jp.playstation.com/scej/online/vcjs10006.html
http://www.jp.playstation.com/scej/online/bcjs30085.html
No longer available after shutdown:
- Viewing and playing other users created stages
- Uploading created stages
- Online play
- Community functions
EDIT:
BTW, Sony has LBP Portable scheduled to be shut down on July 31st and Karting to be shut down on August 31st in NA/EU regions.
https://www.playstation.com/en-gb/legal/gameservers/
https://www.playstation.com/en-ca/games/littlebigplanet-karting-ps3/