In the main game mode, you can unlock items such as a Razor and Soap. What other items can you unlock, how do you unlock them, and what do they do?
Soap - Double points
Razor - removes hair
Duck - Lower penalties
Beer - Slows time, blurs and shakes screen, keeps the kid slowly moving when you stop controlling it
Adoption papers? - Your dad is now signified with an arrow, and is different than your kid would otherwise tell you
Mask - everyone is your dad, but time remaining in the powerup is not shown.
Those are all I have seen, but there could be more.
Well, there is a Halloween Mask which changes the background to a wooden building and all the dads wear costumes; they're all your dads during this.
Beer slows down time and wobbles your screen a bit.