You will be able to get Mocchi, Raiden and Zan sub-breeds from the start of the game, but, you must go through some storyline events to unlock the main breeds at the shrine.

"Available at the beginning of the game" means that if you have a disc that makes the monster, the shrine will allow you to proceed with making it.

Antlan: Available at the beginning of the game.

One Rare Antlan, Dark Antlan (Antlan/?), can only be gotten through combining:
Combine any Chaos Monster (Chaos Ripper, Chaos Dragon or Chaos Durahan) with the Red, Yellow or Blue Antlan. You will get Dark Antlan as the only possible result.

Ape: Available at the beginning of the game.

Baku: Available at the beginning of the game.

Beaklon: Available at the beginning of the game.

Dragon: Available at the beginning of the game.

Ducken: Available at the beginning of the game.

Durahan: Available at the beginning of the game.

Garu: Available at the beginning of the game.

Gitan: Available at the beginning of the game.

Golem: Available at the beginning of the game.

Hare: Available at the beginning of the game.

Henger: Available at the beginning of the game.

Jell: Available at the beginning of the game.

Joker: Available at the beginning of the game.

Lesione: Available at the beginning of the game.

Koropendora: Available at the beginning of the game.

Madillo: Available at the beginning of the game.

Mew: Available at the beginning of the game.

Mocchi: You will be told at the shrine:
"There seems to be something else that's sealing this Saucer Stone. Usually these problems have to do with the breeder's mental state. Is there anything bothering you?"
1) Go to Torble (you must be at B Rank)
2) Talk to the Girl in the Bungle Inn
3) The nurse visits your ranch
4) Go back to the Bungle Inn and get Mocchi Doll
5) Go to the KalaragiJungle
6) Get down to the 6th Floor
7) Fight Geras(for the fight to occur, you must go far enough into an empty room on the 6th floor)
8) Wait a week
9) Phayne's mocchi comes back with the nurse
11) You can now regenerate Mocchis CDs at the shrine.

Mogi: Available at the beginning of the game.

Naga: Available at the beginning of the game.

Ogyo: Available at the beginning of the game.

Pancho: Available at the beginning of the game.

Pixie: Available at the beginning of the game.

Phoenix: 1)go to Kawrea
2)Touch the Gaia Stone
3)Go back to Kawrea after all the commotion
4)Beat Xevion (need to do it two time, for he regenerates) (he's on floor 10)
5)You finish the story line
6)Credits
7)start up game again
8)two weeks after defeated Xevion, you will find a Phoenix feather on your bag
9)Suzaku, the Phoenix shows up and asks you to raise one of her kind

Phoenixes are now able to be regenerated in the shrine.

Plant: Available at the beginning of the game.

Raiden: You will be told at the shrine:
Hmmm...? Hmmmmmm...? A certain level must be achieved before you can raise this monster....Unfortunately you have not reached it. Please try again after you get stronger."

MR4 has a new Breeder Rank up system. While your monsters improve their ranks by beating each other senselessly, you have to do it through the plot. You get Rank D from the first expedition:

1) Go up to the fourth floor. You should see a room that triggers the plot. You'll then leave the dungeon.

2) The Gadget guy will tell you that official tourneys are canned due to the lack of resources. Go back to the cave.

3) Explore it until you hit the 5th floor. You should see the boss that's the big bird monster, Shiden

4) Beat it, and you'll gain a King's Feather. When you leave the dungeon, you should see Cynthie who promotes you to D rank

Rhinoroller: Available at the beginning of the game.

Ripper: Available at the beginning of the game.

Suezo: Available at the beginning of the game.

Suzurin: Available at the beginning of the game.

Tiger: Available at the beginning of the game.

Zan: to unlock Zan:
1)Attain Breeder rank A, and Beat the A class Official Tournament.
2)There will be 3 Promiass adventures, The last adventure ending with the meeting with the Boss, Garp.
3)You will get Rank S .
4)You will be invited to see a tourney match.
5)You will go to Kalaragi Jungle and search for Nazzeni.
6)Defeat "Nazzeni 2".
7)Obtain the Black Egg
Once the Black Egg is yours, you will be able to shrine discs to get Zan.

Zuum: Available at the beginning of the game.

Discovered by: SlaptShotz54, Gizmo the Lemur & Monster Fenrick