14 Continues:

Enter options, go to the Sound Test and play the sounds 1, 1, 2, 4. Press Up twice, and select Sonic & Tails, Sonic Alone or Tails Alone, to start a game with 14 continues. However, the Oil Ocean Zone music will remain forever, and no sound FX will be played.

 

All Emeralds:

In the Select Stage Sound Test, play the sounds 4, 1, 2, 6. Depending of your version of Sonic 2, the emerald getting sound will signal the successfully entered cheat or not. You'll have all the emeralds, and so, the ability to turn into Super Sonic by jumping with 50 rings or more, during the game (and the possibility to see the good endings).

 

Debug Mode:

In the Select Stage Sound Test, play the sounds 1, 9, 9, 2, 1, 1, 2, 4 (1992-11-24? A release date?). A ring will sound when the 4h music starts, if you've done this correctly. Start a stage with A[Start], without releasing A until Sonic appears.
During the game, A and C will make Sonic to jump, and B will change between Sonic mode and object placement mode. In object placement mode, D-Pad will move the cursor (you can fly), A will change the object and C will print it in the stage. You can add almost anything to the stage, including weird objects in some stages, but it will disappear if you go one screen and a half away horizontally. The monitors you can print are teleporters, and thus, will exchange the positions of Sonic & Tails. Use a teleporter in the right place in the Death Egg Zone and you'll get two final bosses fighting you at the same time. The last object before going to the ring again seems to be a big box of messed monitors. Print it and step on top of it to finish the level. If you use the Debug Mode in a 2 Player VS 2D stage, strange things might happen. As in Sonic 1, the Score text will display the offsets for Sonic or the cursor, instead of your score; the number of seconds in Time will be the number of active sprites, and you'll have infinite time. Debug Mode object placement mode doesn't work with Tails.
Note: You can combine All Emeralds and Debug Mode cheats.

 

Scroll A Low Darkening:

If you start a level with C[Start] instead of A[Start], or A+C[Start] for both effects, the Scroll A Low Layer will be darkened. Why you would want to do this, I don't know. Sonic Team used this to see which things are on one layer or another, I believe. BTW, don't release A or C until Sonic appears. You won't see this darkening with Genecyst.

 

Slow Motion:

With the Debug Mode cheat enabled (but not necessarily active for the current level) and the game paused, A will reset the game, C will advance a single frame, and B will put the game in slow motion (you can move and jump with C).