Luigi is hiding in five stages throughout the game. He is in one stage in each of the first five worlds, and when Mario finds him, he can pull him out of hiding through Paperization. When Luigi is found, he will run away and appear in a news article in Decalburg. Additionally, if he is found all five times, he will appear leading the parade in the end credits.
Luigi Sightings
W1-6: Goomba's Fortress
Luigi is sitting on distant wall as Mario crosses a bridge to the central tower before the boss.
W2-5: Drybake Stadium
On the exterior of the fourth floor, Luigi can be seen in a window.
W3-12: Whitecap Beach
As Mario approaches the shore, Luigi can be seen on a distant rock in the background out in the ocean.
W4-5: Whiteout Valley
As Mario rides the ski lift, Luigi will be riding down the opposite direction.
W5-5: Rugged Road
In one of many hidden holes which must be opened by baiting a meteorite onto a soft spot on the ground, Mario will find a hidden hot spring. On the far left of the spring, Luigi can be seen in an adjacent spa.