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 born from Zeus and his aunt and fifth wife, Mnemosyne, who was the personification of memory. The Muses, back then, were simply Deities, in charge of Their own aspects of mortal life. Euterpe was the 'Giver of light', for example. Their function and status as Muses was a later, Roman, addition.
Although most sources say there are nine Muses, the archaic poet Sappho of Lesbos was called the tenth by Plato, a great compliment. It's Sappho's status as 'the tenth Muse' that paved the way to the conventional compliment paid to female poets and those who inspire.
I refer to the muses often; when I'm trying to write but can't find the inspiration, or when I'm trying to sleep but can't because my mind is full of words that want to come out are the most common occurrences. I also thank Them for the inspiration I have gotten to write blog posts or articles.
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:
Homeric Hymn to the Muses:
I will begin with the Muses and Apollon and Zeus. For it is through the Muses and Apollon that there are singers upon the earth and players upon the lyre; but kings are from Zeus. Happy is he whom the Muses love: sweet flows speech from his lips. Hail, children of Zeus! Give honour to my song! And now I will remember you and another song also.
Orphic Hymn to the Muses:
Daughters of Jove, dire-sounding and divine,
Renown'd Pierian, sweetly speaking Nine;
To those whose breasts your sacred furies fire
Much-form'd, the objects of supreme desire:
Sources of blameless virtue to mankind,
Who form to excellence the youthful mind;
Who nurse the soul, and give her to descry
The paths of right with Reason's steady eye.
Commanding queens who lead to sacred light
The intellect refin'd from Error's night;
And to mankind each holy rite disclose,
For mystic knowledge from your nature flows.
Clio, and Erato, who charms the sight,
With thee Euterpe minist'ring delight:
Thalia flourishing, Polymina fam'd,
Melpomene from skill in music nam'd:
Terpischore, Urania heav'nly bright,
With thee who gav'st me to behold the light.
Come, venerable, various, pow'rs divine,
With fav'ring aspect on your mystics shine;
Bring glorious, ardent, lovely, fam'd desire,
And warm my bosom with your sacred fire.
Yet, I strongly feel one honors the Muses best when being creative and inspired. Write, make art, sing, make music, do whatever you can that invites the inspiration of the Muses. Be aware of your inspiration, allow it to flow freely, and if you feel the need, give honor to the Deities who granted it.