Misty is afraid of Bug-types (except those that are deemed cute/pretty), while Iris is afraid of Ice-types (due to them being Super-effective against Dragon-types).
Dawn was afraid of Plusle/Minun because, she was the next caretaker. But, when she hugged them, they shocked her.
Serena has also been shown to be afraid of crossing unstable bridges across high ravines.
Cilan is afraid of the Pokémon Purrloin.
Lillie was afraid of physically touching Pokémon at the beginning due to an incident that happened at Aether Paradise where she was attacked by Nihilego and rescued by Type: Null. Eventually, she overcame this fear and was able to touch them with no problems.
It is known that Sophocles is scared of the dark, but not of the Dark-type.
Also, Clement has a fear of heights.