There are several hidden passageways in this game. Here are a few to get you started.

Level 1-1: To find Level One's secret passage jump down from the second angled ledge and go underground. The first passageway to the left has a false wall which Mickey can pass through. At the end of the passageway there's a variety of bonus goodies.
Level 1-2: Go to the cliff by the forth rope and drop off the left side while holding to the right and you'll get ten apples!
Level 1-4: Jump into the hole by the fifth tree and walk to the left through the rocks to find a room with a free mouse and two stars.
Level 3-1: There's an identical secret passageway, just before you go underground. Proceed just as in Level 1-1 and you'll find another false wall and goody room.
Level 4-3: When you find a tea cup, jump in for a swim and you'll find all types of power-ups. Be careful of the sugar cubes!