Invisible Muncher  [Screen shots: 1, 2]
An invisible Muncher lives in the tall skinny pipe at the end of World 7 - Giant Piranha Plant #2. To reach the top of the pipe, you'll have to fly (P-Wing recommended, as always) to the end of the stage. If you stand on top of the pipe, you'll take damage as if a Muncher were popping up.
--[Found by Deezer]

P-Switch bug  [screen shot]
You can go right through a block containing a P-Switch if you jump twice in rapid succession. The first jump will expose the P-Switch, then the second jump should activate the P-Switch as you go through the block. Worlds 1-2 and 2-3 (as Super Mario) are great places to try this.
--[Found by Deezer]

Airwalking  [Screen shots: 1, 2]
In any level with quicksand, fly as Raccoon or Tanooki Mario (P-Wing recommended), then touch down on quicksand. Fly back up, and Mario will appear to walk through the air as the walking/standing sprite replaces the flying sprite for a short time. During this time, there are no flying sound effects when you press the A button. A cool side effect of this glitch is that you can airwalk behind clouds.
--[Found by David Wonn]

Kuribo's Spiny Shoe  [Screen shots: 1, 2]
With Raccoon/Tanooki Mario in World 5-3, tail-attack a Spiny and pick up his shell. Then, keep holding on to the shell as you bash a Kuribo's Goomba from below to grab the Kuribo's Shoe. You will have Kuribo's Shoe power, but Mario's lower half will be green and black until you lose the shell. Kick the shell away and the shoe will appear normal, or just wait for the Spiny to wake up to lose the shoe.
--[Found by David Wonn]

Brick Blocked  [screen shot]
This is a strange glitch in World 3-6, where Raccoon Mario will temporarily be unable to break a Brick Block. Make sure you're not small at the part with an Ice Block on top of a Donut Lift and a Koopa Troopa on both the high and low platforms. Get to the lower platform, where a Brick Block containing a Super Leaf sits. Quickly kick the Koopa into the block or tail-attack the block to release the Super Leaf, then collect it. If all went well (timing seems to play a part in getting this to work), you should be able to hit the bottom of the Brick Block directly above you several times without breaking it.
--[Found by Deezer]

Super Fire Flower  [Screen shots: 1, 2, 3]
Enter World 5-7 as anything but small Mario and with no Starman power. Bump the first wood block you see to release a Fire Flower, but don't grab it just yet. Hit the first Question Block in the group of three to the right, and the Fire Flower will start to flash. Next, go forward just enough to make the Pile Driver Micro-Goomba attack. Take damage to downgrade to small Mario, then grab the Fire Flower. Instead of transforming to Super Mario, you'll become Fiery Mario!

The block you hit that makes the Fire Flower flash would contain a Starman if you entered the level with Starman power, so something about hitting the block without Starman power while the Fire Flower is exposed causes this glitch. This glitch also works in World 7-3 and with the Super Leaf (though it won't flash) in World 6-1.
--[Sent in by Mario3Lover]

Warp Zone canoe glitch (Japanese version only) [Screen shots: 1, 2, 3]
Hop in the canoe in World 3 and then use a warp whistle. On the Warp Zone map, move to the left and Mario will go off the island! Unfortunately, once you've moved off the island you won't be able to get back on it. You're stuck unless you use another warp whistle, which will place you by the #8 pipe in the warp zone.
--[Sent in by Mega Man 5]

Stuck in a pipe
In Worlds 3-9 and 7-5, there are Ice Blocks right up against a pipe. Kick away some Ice Blocks so that there is one remaining on the right side of the pipe, with one space between the pipe and block. Position "big" Mario underneath that block [1]. Do a ducking jump, and then hold Left (while still holding the Down button) in mid-air. When Mario hits the Ice Block, he'll bounce between the block and the top of the pipe for a short time [2], and then he'll go down the side of the pipe [3]! (If it doesn't work the first time, try again. It shouldn't take too long to do once you know how to set the blocks up.) After the screen scrolls down, Mario won't come down out of the pipe and will appear to have gotten stuck behind the background [4]. From here, you can move left and right to scroll the screen, but there is no way out. If you were Fiery Mario, shoot some fireballs and they will appear to come from different places each time.

The bad news: When time runs out, you'll have to wait a few minutes for the game to return you to the map screen.
--[Sent in by GoldenRoad155]

White block invincibility
In World 5-7, go down the pipe guarded by a fire-spitting Piranha Plant. In the next room is a white block on the ground, with various blocks to the right leading up to a pipe. Duck down on the white block for five game seconds, and Mario will drop behind the scenery. Before Mario returns to the foreground, hurry up to the pipe that leads back up top, and the enemies there won't be able to touch Mario. The invincibility will last until you enter a pipe or exit the stage.

Ice Block magic  [screen shot]
In any level with Buster Beetle, stand next to an Ice Block that Buster is heading for, then press the B Button at the same time that Buster Beetle grabs the block. The one Ice Block turns into two, as both of you end up holding an Ice Block!

Unfiery Bowser
Start the level with a P-Wing and take the lower route in the tall room with Donut Lifts. When you get to Bowser's room, immediately fly up and over the wall on your left. As soon as you get over, drop all the way down, then fly all the way back to the other room. When Bowser appears, he won't be able to breathe fire.

A strange death  [screen shot]
In World 3-9 at the part with the Para-Goomba, have small Mario hit the sixth Block on the right to sprout a Super Mushroom. Let it go past that Block, then bump the Mushroom from underneath again and the Para-Goomba will die.

Strange death #2  [screen shot]
In world 4-3, make your way over to the second instance of those wooden blocks. There should be three brick blocks above you, one containing a 1-Up. Walk over a little more and you'll see a Spiny walking upside-down on some bricks, and a Buzzy Beetle walking on top of the same bricks. When you see this, don't get too close to the Spiny, or it'll drop down. Instead, walk slowly over and stand under the first block of that platform. When the Buzzy Beetle is above you, jump so you kill him as you hit the block. The Spiny will also die, and you didn't even touch it!
-- [Found by Deezer]

Bowser's "secret" room
In World 8-Castle of Koopa, go in the highest door at the end of the room with all the Donut Lifts. You should be in a room with donut lifts, Bowser statues, and at the end Bowser starts shooting fireballs at you even though he's in the next room. Anyway, squat and fly (easiest with a P-Wing) and land on top of the last Bowser statue's head. Stand up and Mario will "moonwalk" through the wall to another room! It's not really a new room; it's the room you can find by taking the lower route after all the Donut Lifts.

Gray Mario  [screen shot]
As Tanooki Mario, press B as you go down a pipe. If your timing is perfect, Mario will turn gray, but not into a statue.

Gray Hammer Mario  [screen shot]
Enter World 7-8 with a Tanooki Suit. Go down the pipe that leads to the secret room that contains a Hammer Brother Suit. Hit the Giant ? Block to sprout the Hammer Brother Suit. Stand in a place where you think it will bounce on you, and press Down and B to turn into a statue. If you get the Hammer Brother Suit while you're in statue form, you will be Hammer Brother Mario, but gray! The bad thing about this cool bug is that you can't go up the pipe at the end; you're stuck in the secret room.

Suicide
In World 5-Fortress, jump up to wedge yourself between the exit pipe and the right side of the screen. Jump a few more times and you will get smashed and lose a life.

Change the color of objects  [Screen shots: 1, 2]
This bug is really hard to explain... Try kicking a Koopa shell into an object just as you scroll that object off the screen. Scroll back and sometimes it has changed color! Usually the object turns green or blue.

Warp to BOOM-BOOM
In World 3-Fortress, go in the third door and press Up again fast, and you'll go straight to BOOM-BOOM.

Invisible Mario
In World 3-9, squat on the white block for 5 seconds. When you drop down behind the background, go down the pipe at the end of the dry land. You won't be able to see Mario in the water.

Wall jump
If you jump at a wall just right, you'll "catch" the wall with your foot, and you'll be able to jump off the wall! This has many practical uses, such as in the beginning of World 6-9. Have small Mario jump off the ice blocks into the wall of ice. When Mario touches the wall, jump again and he should make it over the top. Go right and you'll find the end of the stage!
Note: This glitch requires a perfect jump, and perfect timing.

World 5-1 treasure box glitch (Japanese version only)
Use a P-Wing before entering World 5-1. After coming out the other side of the pipe that leads to the goal, fly up and over the wall on your left. Travel left on the top of the wall, and you'll hear the treasure-box-appearance sound effect. (You are on the ceiling of the bonus area that you can reach by flying straight up at the beginning of the world.) Go back down, and head for the goal. The Nipper Plant under the Roulette Block has transformed into an object that constantly changes from a "5UP" sprite to a Warp Whistle and back again!

Stranger still, after you hit the Roulette Block, a treasure box will appear! If you hit the Roulette Block with an average jump, the treasure box will appear on the left; if you still have the P-Wing, try to time your tail-flaps in the air so that Mario hits the Roulette Block and continues to fly up a little more, which will cause the treasure box to appear on the right because Mario was flying up instead of moving right (like the treasure boxes in the Wandering Hammer Brother battles, this one appears on the side of the screen that is farthest away from Mario). If you get the treasure box to appear on the right, Mario will collect both the Music Box from the treasure box and the card from the Roulette Block, although the graphics will glitch a little and replace the "Course clear! You got a card" message with scrambled background graphics.
(This will only work in the Japanese version because the English version has no tall wall near the goal. See the SMB3: From Japanese to English page for more details.)