Richard's Villa Secret Music
Go into Richard's house and wait until the music has looped a few times. Eventually it will stop, then play a little tune found in Mario Paint and Yoshi's Story, then the regular music will start up again.
This song is called Totaka's Song, as Kazumi Totaka will input this tune into various games he composes.
Create a new game slot with the player name ZELDA (case-sensitive) to hear the game music remixed with alternate instruments.
Beat the game without dying (0 Continues). At the very end you will see a winged Marin fly across the screen, her wish granted. See 0-Death Continue for a way in which to play relatively normally yet still achieve this ending.
Equip bombs to one button and arrows to another, and hit and at the same time. Link will shoot an arrow with a bomb on the end, and it will explode on contact. This is very useful for taking out some enemies.
Just as in all 2D Zelda games, you are invulnerable while using the Hookshot.
Cuccos Attack You
Just like in the previous games, hit any one Cucco around a dozen times and eventually a swarm of invincible flying cuccos will fly around attacking Link.
This will only work for expensive items such as the bow and shovel. Take the item to the counter and choose to buy it, then the moment the text box disappears hit + and save the game. When you reload you should have the item but also have most of your Rupees. This is especially useful for getting the bow and shovel in quick succession, rather than saving up from zero once more. The faster you saved, the more Rupees you will have left.
If you've ever experimented with leaving a shop without paying, you know what happens. But there is still a way to steal items. Pick up any item and walk over to the right. The shopkeeper will turn to face the right-hand side. Then quickly head for the door. If you're fast enough you will get out the door before he turns around and accosts you as per usual.
However when you return to that same shop, the shop-owner will kill you, and your character's name is changed in places like the photo album to "Thief", however your name on the save slot will be the same. There is no way to "clear your name" after having cheated like this.
When you die hit + (+++ in the B&W version) while Link spins around. According to the game's mechanics he is not legally dead, as he still has a chance of being revived at this point by Crazy Tracy's medicine, if he bought it; therefore choosing Save & Quit while he spins will allow you to continue but will not count as a loss. This technique allows you to see the best ending without having to quickly reset upon death.
Kill dogs and cuccos without the fire rod
Go up to the dog or cucco and use the powder on it. This doesn't work in The Legend of Zelda: A Link to the Past.
Kill dogs and cuccos
Once you get the fire rod you can finally successfully slay the dogs and cuccos without retribution.
A notable glitch that effects gameplay is the select glitch. After going past the edge of the screen, but before the new one appears, press . This will warp you at or near the edge of the new screen, depending on when in the transition you pressed it. This glitch can produce new ones, like duplicating items and/or getting them early (which in turn can also cause glitches, such as the "level 0" shield). This can also corrupt your game, though. There have been reports that getting too many heart containers can corrupt the entire game. There are many glitches that grow from this one.
This glitch was fixed in ROM revisions from 1.1 onwards, including all DX versions.
Invisible L-0 Shield
Nintendo patched this glitch, therefore this will only work on the first two revisions of the original, and not on any of the DX revisions.
Start a new game and head for where your sword is. However don't pick it up; instead use the Select Glitch and the sword should have followed you onto this screen (if not return east and try again). Then pick the sword up and return to its true resting-place. There should be a shield there instead. Picking this up downgrades the shield to L-1. It is identical to a normal shield but is invisible. You lose it if Like-Likes eat it or if you get the L-2 Shield.
Explore Color Dungeon on B&W Game Boy
Interestingly enough, it is possible to answer the "What color is my tunic?" question correctly. If you get the answer wrong, just talk to the same Skull Pirate and give the other answer. You can then explore the dungeon, but don't expect to win it without a Color Game Boy or better.
Flying rooster of death
Once you have the Flying Rooster, throw the boomerang and pick the Rooster up. When the boomerang returns it will be unable to disappear back into your inventory because you have not "landed" yet, and so it will circle under your feet. Simply float around in this state and the boomerang will kill anything you touch. However you have to drop the Rooster to change screens so you are unable to go on long rampages. The moment you land after letting go of the rooster this effect will end. 
Early access to Dungeon 8
The fire-spurting is supposed to prevent Link getting through. However after dying and thus using Crazy Tracy's medicine, it is possible to be temporarily invincible and run through the flames. You can then beat Dungeon 8 without finishing Dungeon 7 first.  This can be done anyttime after clearing Dungeon 5.
Weird room in Dungeon 1
Go through the Dungeon normally and when you get to the final boss. Have the Roc's Feather in hand. Fall off the left side of the platform and as soon as you reach the basement. QUICKLY Jump with the Roc's Feather when your at the top of the screen. You will end up in a weird room where there is a bunch of stuff in the way you can't move.