Secret Challenge 01 (Credit: JonLacksAnH)
Area 01: Once you reach the cave, drop to the bottom of the shaft and head left. Kill both the guards and robot laser guard, then continue left taking out the guards an laster-bots until you're forced to go up. On going up, take the first left you can. Blow away the wall in your way to open the door to the first secret challeng room. Secret Challenge 02 (Credit: JonLacksAnH)
Area 02: Drop down and go left from the entrance to the Boss' Room. Keep heading left and up until you reach a fork that goes both up and to the right, and up and to the left. Go up each path and pull down the respective switches. After that, head to the right to the next elevator. Take it to the top and go in the door on your left for Secret Challenge 02. Secret Challenge 03 (Credit: JonLacksAnH)
There are alternates paths to this level depending on whether you go to it before or after you complete the game.

Area 03: (Before completeing the game) Go to the ledge just below the mountain summit and head right to the dead end. Equip the bazooka, hook on to the ceiling, and fire. This should reveal a small tunnel. Ignore it and head back to the bottom right of the mountain. As you make you way down, a tone should sound saying the secret area has opened, and you should find a door waiting for you at the base of the mountain.

(After): Go to the ledge just below the mountain summit and head right to the dead end. Equip the bazooka, hook onto the platform, and fire a rocket up to the ceiling to your right. This should reveal a small tunnel. Ignore it and head back to the bottom right of the mountain. As you make you way down, a tone should sound saying the secret area has opened, and you should find a door waiting for you at the base of the mountain Secret Challenge 04 (Credit: JonLacksAnH)
Area 04: On starting the level, swing past the spikes then head all the way up and left. You'll find a cracked wall that can be opened with your bazooka or grenade. Head in and go left over the spikes. Next, pull yourself upward through the colored platforms and then go traverse the spikes to your right. Below the spot where you find the Yashichi is a wall that can be blown open with grenades. Behind its hidden door is Secret Challenge 04. Secret Challenge 05 (Credit: JonLacksAnH)
Area 05: Go to the top of the level and shoot the blue block there. After that, head back down to the communications room. Go down one floor and to the left you'll find a door that was previously closed. Inside is Secret Challenge 05. Secret Challenge 06 (Credit: JonLacksAnH)
Area 07: Go to the communications room and to the right should be four lampposts. Swing over the farthest right lamppost, swing back to the left and grapple upwards. Use a shotgun blast to give yourself swinging momentum again then make your way to the door you'll see. This door houses Secret Challenge 06. Secret Challenge 07 (Credit: JonLacksAnH)
Area 11: From the communication room, go left, clear the block in your way, and try to land just before the first long drop. Drop down and while falling, you should see an aclove. Grapple the ceiling of the space to swing inside then go up to the hole in the wall to your left. Shoot your grapple arm inside (forward + B) to grab a bolt inside then hit B to loosen it. Continue on down the shaft and go back to the communications room. Again, you head left over the block, but this time you need to grab one of the round objects floating in the air. Use it to swing yourself past the shaft you fell down earlier to a second shaft. Again, while falling, you'll see an alcove to your left Grapple onto the ceiling and swing again to find a room leading to Challenge 07.