Guide: how to tame animals?
1) Find which animal you want to tame.
2) See here whether it is tamed or not (for convenience, use F3 for a quick search.)
3) Finish off the animal almost to the end, so that there is a thin stripe and choose Tame, click the mouse several times on the animal until the chat says "Tamed *name of the animal*".

Wyverns:
Angel Dragon - tame
Imperial Dragon - tame
Dead Forest Dragon - tame
Purple Crystal Dragon - tame
Tide Dragon - tame
Blue Neon Dragon - tame
Zombie Dragon - tame
Rainbow Dragon - tame
Silver Dragon - untamed
Toxic Dragon - untamed
Candy Dragon - untamed
Aqua Dragon - untamed
Forest Dragon - untamed
Devil Dragon - untamed
DeadFire Dragon - untamed

Dragon Monsters:
Magma Beast - untamed
Lava Beast - untamed

Other:
Golden Beetle - tame
Phoenix - untamed
DeadFire Frog - untamed
DeathSpeaker - untamed