Wyverns spawn at any lateral map location, day or night, but generally at Floating Island altitudes. They will follow descending players down to ground level, and can fly through blocks, but tend not to fly below ground unless they are Arch Wyverns. A Wyvern will not spawn if another one is already alive, and like most enemies able to pass through tiles, it will not be able to spawn if the player themselves stays in front of player-placed walls, even if there are valid spawn points for the Wyvern. Outside of Crimson, Corruption, and Dungeon biomes, and except during events, having at least one NPC near the player will also prevent them from spawning.
It is possible to kill a Wyvern while its head is off the map. As the head is the part dropping the loot, this can result in the loss of the loot.
Wyverns are a total of 14 segments long. The first segment is the head, the second and seventh are the legs, the third through sixth and eighth through eleventh are body segments, and the last three are the tail segments.
Tips
Killing Wyverns can be hard for new Hardmode players, so its recommended to stay away from them until better armor and weapons can be obtained.
As with any multi-segmented enemy, piercing and explosive weapons can strike multiple segments at once, dealing several times their normal damage.
The head of a Wyvern has lower defense than its tail or body segments, making it more vulnerable to damage as it charges the player. However, the head also deals twice the damage as the rest of the body on contact, making it risky to face a Wyvern head on.
It is easier to fight a Wyvern in Space, as the low gravity helps to dodge its attacks.
On the 3DS version, Wyverns do not despawn when all players die, unlike most other enemies/bosses.
Engaging a Wyvern with a Gravitation Potion in Space and allowing yourself to free-fall whilst moving horizontally can put you at almost the same speed as the Wyvern. With adequate movement speed buffs you can match or even outpace it, and while the Wyvern is tailing you, attacking it with piercing weapons such as the Golden Shower can make short work of it, even in Expert mode.
Trivia
Although the name "Wyvern" refers to a type of western dragon depicted as having two wings and two hind legs, but no arms or forelegs, Terrarian Wyverns are shaped after Chinese dragons in Chinese folklore.