Pokeballs are often found in Gift Bags and Smiling Gift Bags. Some are Expert mode and Fun Mode exclusive.
(Incomplete! Will continue editing :D)
#83: Farfetch'd[]
"farfetch'd will follow you"
'it seems this pokeball smells like onion'
Can be found in a Smiling Gift Bag. This Pokeball is Expert Mode exclusive.
#92: Ghastly[]
"I hope you don't get poisoned"
'You feel one of the ghost type pokemon is inside of this'
Can be found in a Gift Bag.
#143: Snorlax[]
"Snorlax will follow you"
'You hear snoring sound'
Can be found in a Gift Bag.
#172: Pichu[]
Rare drop from Plantera
Fun mode-only
#225: Delibird[]
Rare drop from Santa Claus NPC
#258: Mudkip[]
An item from boss hunter merchant's shop
#281: Kirlia[]
Random drop from smiling gift bag
#303: Mawile[]
Random drop from smiling gift bag
#321: Wailord[]
Random drop from normal gift bag
Masterballs[]
#150:Mewtwo[]
Random drop from normal gift bag
250[]
Random drop from normal gift bag
Ho-Oh pet
251[]
Random drop from normal gift bag
Celebi pet
382[]
An item from boss hunter merchant's shop
Kyogre pet