There are a total of 12 hidden stages in the game with 8 of them appearing on the world map automatically when you have collected ALL the Red Diamonds in a world. Listed below are the remaining 4 which will appear when you complete the corresponding task.

Unlock Stage 9-1:
Complete the game without using any Continues.

Unlock Stage 9-2:
Complete stage 9-1.

Unlock Stage 10:
Collect ALL the Red Diamonds in a world to unlock stage 10 in each world.

Unlock Stage X-1:
Use the warp block which is found directly above the starting position in stage 4-3 (use bubble gum to reach it).

Unlock Stage X-2:
Go to the first checkpoint flower in stage 6-7 and then continue to the right until you find a Bazooka Bee. Then bounce off it's back to find the warp block to stage X-2.