Well these are the abilities that have no effect at all in a trainer battle:
Honey Gather (May gather Honey after a battle)
Pick Up (May pick up items)
Stench (Helps to repel wild Pokémon) Gen I to Gen IV
Illuminate (Raises the chance of meeting wild Pokémon)
Run Away (Enables a sure getaway from wild Pokémon)
Hope I helped :)