It seems everyone's favorite Pokemon is just looking for a photo op. Here's some locations for some very special photographs.

To see Pikachu rolling on a ball go to the Tunnel. When you start the stage take pictures of the Pikachu immediately. He should move farther down. Keep taking his picture. He will stop and jump on a rolling Electrode.
To see Speed Pikachu go to the River. At the very end on your left you should see Pikachu hiding behind a tree. Throw an apple at Pikachu and it should fall down. Then he will run back and forth as the Speed Pikachu.
To see Pikachu surfing go to the Beach. Throw apples at the first Pikachu you see. Throw the apples on the right side of your tracks. There will be a surf board there so try to lead him there. If you do it right he will surf.
To see Pikachu on a stump go to the Beach stage. The first time you see Meowth chasing a pidgey out of the long grass, throw pester balls into the long grass to get Scyther. After you get Scyther, 2 Pikachus should come out to the stump on your left side. Lead them with apples and play the poke flute.
In the cave, towards the end, you will see a zubat carrying Pkachu. Hit the zubat with apples and pesterballs until he drops Pikachu. Pikachu will sprout balloons and fly away.
Save Pikachu from the Zubat in the Cave and awaken Articuno from his egg by playing the Pokeflute. When you get to the end of the level next to where Jigglypuff's stage is, turn around and you'll see Pikachu riding on Articuno.