The Muses (Μοῦσαι) are either three or nine in number, depending on the source. Plutarch, in his Quaestiones Conviviviales, named three Muses; Melete (Practice), Mneme (Memory), and Aoede (Song), but Hesiod described nine of them in his Theogony; Calliope (epic poetry), Clio (history), Euterpe (flutes and lyric poetry), Thaleia/Thalia (comedy and pastoral poetry), Melpomene (tragedy), Terpsikhore (dance), Erato (love poetry), Polyhymnia/Polymnia (sacred poetry) and Urania (astronomy). Due to the influence of Hesiod, Hómēros and others of their time, it's the nine Muses we now go with.
These nine muses were most often considered born from Zeus and his aunt and fifth wife, Mnemosyne, who was the personification of memory. Their parents are of importance and there is logic behind it: Zeus is the King of the Gods; He rules and inspires all. From Him, the Muses get Their influence over life and it shows how important art in its many forms was to the ancient Hellenes.
Perhaps more telling, however, is who Their mother is. Mnemosyne is a very important and influential Goddess who features heavily in death rites and in the eleusinian Mysteries, especially. Those initiated into the Mysteries would get to drink from Her water once dead and not forget their life like all others would. In the land of the living, memory is of great importance as well: being remembered meant to live forever. Many ancestral rites focussed on the recounting of names and deeds to accomplist that goal and this was done though poetry and song, often accompanied by music and dance. Without Her daughters, Mnemosyne could not fullfil Her primary task: to help mankind remember the honoured dead.
The muses are well represented in both mythology as Hellenic art. Not only does nearly every hero, poet and even some of the Theoi call out for Them when They're in a bind, but there is even a tragic story in which nine young women get turned into birds for their hubris. In this myth, King Pierus, king of Macedon named his nine beautiful and talented daughters after the muses and went on to boast that the Pierides--his daughters--were equal or even better in their arts than the Muses ever were. Needless to say, neither the Muses, nor the Gods took to this kindly. As punishment for his hubris, Pierus had to watch as his beautiful daughters were transformed into Magpies.
Apollon Mousagetēs, an epithet of Apollon, is said to lead the Muses. His name means 'Apollon Muse-leader'. As such, he can be seen depicted on vases and murals with the Muses.
Here is Apollon to the left, followed by the Muses. If I tell you Calliope carries a writing tablet, Clio carries a scroll and books, Erato is often seen with a lyre and/or a crown of roses, Euterpe carries a flute, Melpomene is often seen with a tragic mask, Polyhymnia is often seen with a pensive expression, Terpsichore is often seen dancing and carrying a lyre, Thaleia is often seen with a comic mask and Urania carries a pair of compasses and the celestial globe, how many can you pick out?
Within Hellenismos, worship, libations and offerings to the Muses is an individual choice. If you feel a need to do so--and poets, musicians, writers and other artists might definitely feel that need--do so. I know I do. As did the ancient writers; there is both an Homeric and an Orphic Hymn to the Muses one can use to honor Them. Most often, however, They will be called to in times of need--and that need can be anything, from giving a funeral orration to presenting a rapport in school. Any time when you need to perform your best, by word or instrument or dance, pray and sacrifice to the Muses and hope They hear you.
