The Minotaur and the Labyrinth
Hercules' Labors
Persephone & Hades
King Arthur
Gawain and the Green Knight
Myths are based in oral tradition, which means that the stories were shared person to person, without the popularization of a written text. While the most well known myths ended up in written history, there is almost certainly myths that didn't get added that are currently lost.
Variations in myths come from their basis in oral history before being written down. When these tales were passed to the next person, slight alterations happened. These changes were either because of a difference in opinion or were the fault of human error. But as soon as the myth got transcribed into written history it maintained a semi-solid state. Now the major changing force in old myths are other old records of myths that either show a different view or contain more information.
With the sordid tales of some mythologies, it can bring into question whether the people believed in the myths or just passed them down as part of their tradition. To give a measure of proof, some mythologies ask for shows of faith or ways to practice said belief. From visiting shrines in Shinto, to Writing protective glyphs on burial sites in Norse and Egyptian tombs, There are physical monuments and artifacts that back up the faith people have.
There should be a caution message for anybody interested in researching mythology. Mythology is not really meant for kids, though it can be made palatable to a younger audience. Most myths include some form of suicide, sexual themes, sexual assault, incestuous relationships, and/or beastiality. These themes are not suitable for younger audiences and should be prefaced with a warning such as this.
Greek Myth: Hesiod's Theogony, Ovid's Metamorphoses, Homer's Iliad & Odyssey
Celtic Myth: The Ulster Cycle, Mythological Cycle, Fenian Cycle, and the Cycle of the Kings
Egyptian Myth: The Pyramid Texts, and Book of the Dead.
Norse Myth: The Prose and Poetic Eddas
Christianity: The Bible, not mentioning the apocryphal texts
Aztec Myth: Codices, and Spanish textual records
Shinto: Kojiki and the Nihon shoki.
Islam: Housed in the Quran and Sunnah
Judaism: The full Bible and the Talmuds
Hindu Myth: The Mahabharata and Ramayana
Mesopotamian Myth: Clay tablets found from excavations.
Roman Myth: The Aeneid of Virgil, Livy's history, and Dionysius's Roman Antiquities