In Resident Evil: Operation Raccoon City for the PS3 and Xbox 360, purchasing an ability does not automatically mean you can use it.
All passive abilities are effective all the time, but you can only have one active ability to use at a time.
On the character selection screen, if your abilities box says “none” then you have no active abilities selected.
Go into the abilities, and highlight one of the active abilities. On the Xbox 360, press A to select that ability, and on the PS3 press X to select that ability.
Make sure to read the text of the ability carefully to see how it is used in battle.