Jump around
In #4 Ludwig's Castle, run past the falling spiked ceiling and wait by the door. After the ceiling hits the ground, the ceiling-hitting-the-ground sound effect actually replaces the Spin Jump sound effect until you exit the screen via the door.
-- [Found by nintendoexpert89]

Invisible object  [Screen shots: 1, 2, 3]
There are three known "invisible objects" that, when Yoshi touches with his tongue, will make a thumping noise like when Yoshi tries to eat inedible stuff like keyholes.

• Yoshi's Island 3: At the first orange pipe, stand somewhere under the two coins. Face left, and hold Up as you shoot out Yoshi's tongue. The invisible object is somewhere below the coins and above the pit.
• Yoshi's Island 3: At the last two "swing lifts," stand a little to the left on the first of the two lifts. As it swings down, aim Yoshi's tongue about four blocks directly to the left of the second lift's pivot point.
• Top Secret Area: Stand somewhere under the two ? Blocks on the left. Face right, and hold Up as you shoot out Yoshi's tongue. The invisible object is somewhere below and to the left of the middle ? Block.

-- [No. 2 found by Grey Mario]

Freeze the game
Bring Yoshi into a level with Berries, and you must be either Caped Mario with a Fire Flower in reserve, or Fiery Mario with a Cape Feather in reserve. Position Yoshi's mouth directly underneath a Berry, and then press Select to drop your item in reserve. Time a jump so that Yoshi's mouth eats the Berry as Mario collects the item in reserve, and the game will freeze up. The only way out is to reset the SNES, or press Start then Select to exit the level.
There is a cool variation of this glitch that not only freezes the screen, but increases your Coin total forever, eventually giving you 99 extra lives. This seems to happen rarely, and when the item in reserve is collected a fraction of a second after Yoshi's mouth touches the Berry. A good place to practice this is in Yoshi's Island 2.

Killer corner
Bring small Mario into Chocolate Island 4. When you start the level, immediately turn around and walk the short distance to the left wall. Keep facing left, and then release Left the Control Pad, letting Mario slowly slide down the slope a little. Then press B to jump, and try to hit Mario's back on that corner sticking out of the ceiling. If you hit it just right, Mario will get killed.
-- [Sent in by Juan Pablo Peloso]

Anti-gravity room  [Screen shots: 1, 2]
Enter Vanilla Dome 2. Make sure you are Fiery Mario (keep a Cape Feather in reserve for more fun) by the time you bring the first P-Switch over to your left, where you'll run through a thin wall of Coins and into a bunch of brown blocks. Stomp the P-Switch, and collect some Coins from the bottom row (run and duck). Keep ducking until the P-Switch's power stops and the Coins change back to brown blocks. Then, spin jump until you drop down into the wall. Inside this wall is particularly special because any object inside that isn't touching Mario will float from side to side. To see what I mean, shoot some fireballs; they won't bounce... instead they shoot straight to the other side! With careful planning you can take Yoshi with you, and even the P-Switch. Leave them above you inside the wall and watch them float back and forth.
-- [Sent in by Juan Pablo Peloso]

Play that funky music...
Bring Yoshi to Funky. Wait for the timer to drop under 100, then when the music speeds up, eat a Green Berry to take the timer back over 100. When the timer drops under 100 again, the music will speed up again! Repeat this process, and see how fast you can get the music to go.

Magic Mouth #1
Bring Yoshi and Caped Mario into Donut Secret 2. At the part near the end with the lone Koopa Troopa and 2-3 Spinys, have Yoshi eat the Koopa, then fly up above the pipe next to the wall of red Winged Koopa Troopas. Make sure you can see the goal, then spit out the shell. You'll see the "shell" is really a cycling conglomeration of stuff. To get a better look, spit it inside the pipe that hangs down. Anyway, when you spit it out, the goal tape will disappear, and you can walk through the goal without clearing the course. To get the tape back, fly over the pipe and come back.
-- [Sent in by Chomp]

Magic Mouth #2
Make sure you have the Green and Red Switch Palaces cleared, and bring Yoshi and Caped Mario into Star World 4. Get all the way to the Key, and have Yoshi eat it. Run and fly to your left, hitting all the Green [!] Blocks. Get all the feathers, then get up on the platform with the Koopa that kicks a shell. Spit out the Key and it you'll see that it turned into a deformed spinning thing. If you spit it between the two pipes, you can experiment a little. You'll notice that Mario cannot stomp it, nor can it hurt Mario or enemies. But if you make Yoshi eat it, you'll get a Cape Feather.
-- [Sent in by Chomp]

Double Yoshi
Chocolate Island 5 is the best area for this, but it can be done anywhere with similar conditions. Start off with Caped Mario without Yoshi. Fly up from the start to get the Gray P-Switch and put it down on top of the blue P-Switch. If you have it in the right spot you can pick both up at once. Now, throw them both up at the block containing Yoshi. You'll notice Yoshi will sprout out quicker than usual. Mount Yoshi, then leave him on the blocks below. Go back up top and jump as if Yoshi had just sprouted out. You'll hear the zip-zoo sound, but you'll be walking around like normal. Hit the P-Switch to turn the blocks into coins, and Yoshi will fall down the pit. The second Yoshi will appear where Mario is standing!
-- [Found by David Wonn]

The magic Coin
In Donut Secret 2, have Yoshi spit a shell at the first ? Block. A faster music theme should play as a line of Coins trickles from the block. Eat the Coin that moves, and now the faster music will stay on until you beat the stage.

Walk through walls
In Valley of Bowser 2, make your way to the maze with the rising and falling floor. Get under an overhang of some sort, and duck. When the floor starts to rise, keep ducking until just before you're about to get smashed, then spin jump. If all goes well, Mario will go through the wall! I'm not sure if this works with small Mario yet.

Buns of steel
Sliding down on these enemies will result in some really weird symbols after you defeat them... Though it's just a graphics glitch, it looks like Mario totally smashed that enemy!

• Iggy and Larry Koopa's fireballs (shell-in-the-lava bosses) - Let them chuck a fireball at you, then slide into it.
• Grinder - In Forest Fortress, slide down the Triangular Blocks into the Grinders.

Pipe glitch
Leave something on top of the pipes that go up and down, and it will end up inside the pipe as it extends. Experiment with P-Switches, Shells, Yoshi, etc.

Confused Spike Top
In Vanilla Dome 1, jump over the first Spike Top, then follow him back to the left. When he gets to the end, he will climb around the very left edge.

Everlasting Grab Block
In Valley of Bowser 2, take a Grab Block through the pipe to the Goal, and it will never "melt" in your hands.

Everlasting cloud ride
Shoot a fireball at a Lakitu that's riding in a cloud, and make sure you get the Coin! Now, jump in his cloud, and it will last forever.

Gray Buzzy Beetle shell
In Donut Plains 2, carry a Buzzy Beetle shell through the pipe to the Goal. When you come out, the shell has turned gray, and the beetle inside won't come out!

Color-changing pipes
In Donut Secret 1, swim up the first pipe to a room with coins and a P-Balloon. Notice the pipes in that room are orange. Now get the P-Balloon, and float as far up as you can. When you come back down, the pipes are now green! For a variation, get the P-Balloon then float all the way over to the right (just to make this easier) then float up. If you don't go any farther up than two or three block heighths below the group of two Coins close to the wall, when you come back down, the bottom of the pipes are green, and the rest is orange! You can change one-third to two-thirds of the pipe green depending on how high up you float.

Clogged chimney
Go to Yoshi's House and just keep pressing the L and R buttons. Sometimes the birds will pause, and the smoke from the fireplace will skip. If you're lucky, the smoke will then spaz out. It's hard to describe but you'll know it when you see it. Re-enter Yoshi's House if you can't get it to work after a minute or so.

Magic fireballs
At the start of #2 Morton's Castle, go past the Thwimps and you will get to an escalator. Shoot fireballs on and around the spinning blocks at either end of the escalator. If all goes well, the fireballs will teleport above you. It's hard to explain so just try it! This works with the escalator in #7 Larry's Castle also.

9,999,990  [Screen shot]
Get a cape and find a bunch of Wigglers close together. Hold Y and B as you jump on them one after the other without touching the ground. As usual, you will get more and more points for each one you stomp, then eventually 1UP, 2UP, 3UP. Each time after 3UP you will get a bunch of weird symbols, some coins, and a TON of points. The last time I did this I went from 6,714,540 to 7,247,020 to 7,799,980 to 7,912,620 points with each jump.
Ideal course: Forest of Illusion 1. A bit after the Midway Gate, there are floating logs with 3 Wigglers roaming across them. Jump on them and keep scrolling back and forth so they change from mad (red) to normal so you can stomp them again.

Keyhole craziness
Jump off Yoshi to an area where there is a Key and keyhole above Mario and not visible on your screen. Make sure the screen does not scroll up to the keyhole. Take the Key and insert it into the keyhole. The keyhole will expand from the center of the screen instead of from the actual keyhole.
Ideal course: Donut Plains 1.

Glitchity graphics
In Chocolate Island 3, bring Yoshi to the first Goal. Get on the block that sprouts the vine, jump up, then jump off Yoshi. Just like the 'misplaced keyhole' bug above, you want to get on this platform without the screen scrolling up. Now go through the goal. When the course clear spotlight thingie focuses in on the bottom of the screen, your screen will flicker. Sometimes the screen will turn black and the game will freeze up. If it doesn't, you'll go back to the map screen like normal... except the colors will be all messed up. Next time you go in a level, the colors will be back to normal, but you'll have several black vertical lines in the background. Reset the SNES to get the game back to normal.
WARNING: One time, when the screen completely froze, some weird music kept looping, and I had to reset the SNES. When the game came back, all my saved games were erased! Who knows what else can happen...

Block factory  [Screen shots: 1, 2]
I still do not fully understand this bug, and it's really hard to explain. It usually happens when there's a bunch of blocks around you. Pick up an item like a P-Switch or something, and jump. Try to kick the item into a block and hit your head on a block in rapid succession (or possibly even simultaneously). Sometimes a block will appear out of nowhere! The type of block you can create depends on your location.
Ideal course: Chocolate Island 5. Pick up the P-Switch and use it as your item with the square of blocks above you. You can create lots of Rotating Blocks.
In Donut Plains 1, take the key and use it as your item with the wall of Green [!] Blocks. You can create a few Green [!] Blocks here.
-- [Found by Deezer]

Smashola
Find an area with lots of Rotating Blocks. Get a cape, and spin jump while rapidly pressing Y and either Left or Right on the Control Pad. Sometimes you will travel really fast through the blocks and end up getting smashed against a wall, and in certain situations you will even end up in a wall.
Ideal course: Star World 1.

Magic Yoshi
Get hit by an enemy the same instant you hit a block that contains Yoshi. Jump up to mount Yoshi, and Mario will bounce in mid-air above him, and then land on him. For each bounce, the mounting-Yoshi "Zip-zoo" sound effect will play, creating a stuttering sound effect like: "Z-z-z-z-z-zip-zoo". But wait, there's more! Throw Yoshi down a pit, then go back to where you got Yoshi. He should be there again!
The place where I discovered this was in Donut Plains 1, where you have the Super Koopa flying at you as you pass a [?] Block that contains Yoshi. You may have to use your cape to hit the block as you get hit by the enemy.
-- [Found by Deezer]

Displaced object
If you have Yoshi, sometimes if you're eating Berries and moving/turning around erratically, you will leave a Berry motionless in mid-air. It's pretty weird and rather hard to explain.
Ideal course: Yoshi's House.
If you have Baby Yoshi, sometimes when you're eating your fifth enemy for Yoshi to become "big", you'll eat the enemy but leave him behind as well. The enemy drops to the "background" and you can't stomp it or touch it, and the enemy cannot affect you as well. Again, rather hard to explain.
Ideal course: Star World 4.