When i go to Pet Trainer, I can’t move my pet to the stable. What is the secret? Does it have to be (can’t think of word) put away-? which makes it lose 50% health. 
You first need to buy a slot from the stablemaster. There’s 2 slots to buy. Once you get the first one, just drag your pet into the slot to stable it.
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Oh, I thought two slots were free.
This is probably true in battle for azeroth but not in classic.