It was not until Hellenistic times that the following systematic set of functions became associated with them, and even then some variation persisted both in their names and in their attributes:

According to Pausanias, who wrote in the later second century AD, there were originally three Muses, worshipped on Mount Helicon in Boeotia: Aoide (song or tune), Melete (practice or occasion), and Mneme (memory). Together, these three form the complete picture of the preconditions of poetic art in cult practice. In Delphi too three Muses were worshiped, but with other names: Nete, Mese, and Hypate, which are assigned as the names of the three chords of the ancient musical instrument, the lyre.Alternatively, later they were called Cephisso, Apollonis, and Borysthenis - names which characterize them as daughters of Apollo.A later tradition recognized a set of four Muses: Thelxinoë, Aoide, Archē, and Melete, said to be daughters of Zeus and Plusia or of Ouranos. One of the people frequently associated with the Muses was Pierus. By some he was called the father (by a Pimpleian nymph, called Antiope by Cicero) of a total of seven Muses, called Neilṓ (Νειλώ), Tritṓnē (Τριτώνη), Asōpṓ (Ἀσωπώ), Heptápora (Ἑπτάπορα), Achelōís, Tipoplṓ (Τιποπλώ), and Rhodía (Ῥοδία). Mythology
According to Hesiods Theogony (seventh century BC), they were daughters of Zeus, king of the gods, and Mnemosyne, Titan goddess of memory.
Here again, Calliope (epic poetry) carries a writing tablet; Clio (history) carries a scroll and books; Euterpe (song and elegiac poetry) carries a flute, the aulos; Erato (lyric poetry) is often seen with a lyre and a crown of roses; Melpomene (tragedy) is often seen with a tragic mask; Polyhymnia (sacred poetry) is often seen with a pensive expression; Terpsichore (choral dance and song) is often seen dancing and carrying a lyre; Thalia (comedy) is often seen with a comic mask; and Urania (astronomy) carries a pair of compasses and the celestial globe. Functions
In society
The Greek word mousa is a common noun as well as a type of goddess: it literally means art or poetry. According to Pindar, to "carry a mousa" is to excel in the arts. The word derives from the Indo-European root *men-, which is also the source of Greek Mnemosyne and mania, English mind, mental and monitor, Sanskrit mantra and Avestan Mazda. The Muses, therefore, were both the embodiments and sponsors of performed metrical speech: mousike (whence the English term music) was just "one of the arts of the Muses". Others included science, geography, mathematics, philosophy, and especially art, drama, and inspiration. In the archaic period, before the widespread availability of books (scrolls), this included nearly all of learning. The first Greek book on astronomy, by Thales, took the form of dactylic hexameters, as did many works of pre-Socratic philosophy. Both Plato and the Pythagoreans explicitly included philosophy as a sub-species of mousike.