It depends on the monster, really. Most of them just died, like any other creature would. Some of them, however, were placed in the heavens because they were either favored by one of the gods, or held some sort of specific significance (like Cancer - Kankeros - created by Hera to prevent Heracles from slaying the Hydra, but was crushed underfoot)
The cookie monster
any of the sea serpents
He did what had to be done to help his brothers to defeat the enemies!
In Greek mythology, it wasn't the weapon, but the person who was important. Not just anyone could slay a chthonic monster. You had to be special, and generally that meant divine parentage of some kind.
Theseus slayed the minotaur, who was half man, half bull.
slaying in Monster Hunter (MH) means you kill the boss/biggest/required monster in the quest using your weapon. after that, you can "carve" for its parts by pressing "O"
Sea Beast is the name of the water monster in Beowulf. He had to slay these minor creature on his way to slay Grendel's mother in her lair.
Though she was not exactly a 'beast', Perseus was known for slaying a monster, the gorgon Medusa.
You climb on the ropes to get to the back of the monster and get passed the spike ball after you do that you pull the lever and the monster will fall asleep and and you stab the monster in the mouth and do this twice more and you will defeat him.
She helped slay a Minotaur and was granted immortality by Artemis.
slay apceros and carve near there shells (apceros are the herbivores found in the desert) also they are easily carved from congalalas
This verb in Greek language is ΦΟΝΕΥΩ / φονεύω (phonevo), or the most commonly in use: ΣΚΟΤΩΝΩ / σκοτώνω (scotono).