Apollon has at least two epithets that refer to Him as the leader of the muses. These are Μούσαρχος (Mousarkhos) and Μουσηγετης (Musagetes). The latter has been translated as patron and companion of the muses as well as leader of the muses.
There are actually three different sets of muses, two of which bear a relation to Apollon. The first set is the set that we think of most often when we think about the muses. They are the Olympian muses, the set of nine muses who are the daughters of Zeus and Mnemosyne, the goddesses of music, dance, and knowledge. The lyre (kithara) and flute have become symbols for all nine of the muses, and the swan and magpie are both sacred to Them. In art, they are sometimes depicted wearing crowns of feathers, which refer to their triumph in musical contest over the Sirens (who lost their feathers to the muses). The muses have sometimes been thought of as nymphs, which would affect the offerings given to them as many do not give wine to nymphs, but you will have to develop your own relationship to Them and rely on your own UPG (unverified personal gnosis) for that information. I have offered wine to Euterpe before with no sense or indication that it was taboo or not accepted.
Καλλιοπη – Kalliope (Calliope) is the muse of epic poetry. Her attributes include a tablet, stylus, and roll of paper/scroll.
Κλειω – Kleio (Clio) is the muse of history. Her attributes include a scroll and an open chest of books.
Ερατω – Erato is the muse of erotic poetry. Her attributes include the lyre.
Ευτερπη – Euterpe is the muse of lyric poetry. Her attributes include a flute.
Μελπομενη – Melpomene is the muse
of tragedy. Her attributes include a tragic mask, the club of Heracles, a sword, a crown of vine leaves, and a pair of cothornus (cothornoi?) – they are calf-length boots with upturned toes, which were used to costume characters in tragic plays.
Ουρανιη – Ouranie (Urania) is the muse of astronomy. Her attributes include the staff and the globe.
Πολυμνια – Polymnia (Polyhymnia) is the muse of religious hymns. She often appears in art without attributes to identify Her. Her posture is usually one of contemplation.
Τερψιχορη – Terpsichore is the muse of choral song and dance. Her attributes include the lyre and plectrum (pick).
Θαλεια – Thaleia (Thalia) is the muse of comedy. Her attributes include the comic mask, a shepherd’s staff, and a wreath of ivy.
The second set of muses is a set of three to five, depending on your source and view. When They are a set of three, it is Arche and Thelxinoe who are generally excluded. They are titans, daughters of Ouranos and Gaia. They are referred to as the elder muses or titan muses.
Μελετη – Melete is the muse of practice.
Μνημη – Mneme is the muse of memory and may very well actually be Mnemosyne, the mother of the previously mentioned set of nine muses.
Αοιδη – Aoide is the muse of song.
Αρχη – Arche is the muse of beginning.
Θελξινοη – Thelxinoe is the muse of charming the mind.
The final set of three muses are the Delphic muses. They are the daughters of Apollon. They are referred to as the younger muses and the Apollonides. There are two conflicting sets of names for this set of muses and nothing (that I’ve found) is known of their domains separate from from what can be inferred by their names. I am not certain of any correlations between the two sets of names, so I am not intending to imply that Kephiso is equivalent to Nete, nor the same for Apollonis and Mete, nor Borysthenis and Hypate.
Κηφισω – Kephiso was named after a local river.
Απολλωνισ – Apollonis means daughter of Apollon.
Βορυσθενισ – Borysthenis means strength.
Νητη – Nete refers to the lowest note on the lyre.
Μηση – Mete refers to the middle note on the lyre.
Υπατη – Hypate refers to the highest note on the lyre.
A note about the lyre: The number of strings on a lyre and the notes which they play vary. Some sources talk about seven stringed lyres because the number seven is sacred to Apollon, but in art, the number of strings shown varies. The lyre that I play has 13 strings. I chose it because I wanted more potential for what I could play. The more strings one has, the more notes one has, the more songs one can potentially play. A key thing to consider when choosing music for the lyre (excepting a few modern versions, but not mine) is that one can tune a string up or down a bit, but this cannot be done in the midst of playing a song, so if you want a string to play a sharp or a flat, the note must be sharp or flat throughout the song and then the lyre must be retuned before a song with different notes. All of this is to say that on my lyre, the lowest note is usually a C, but it can be tuned down to a B flat if necessary. The highest note is an A, which can be tuned up to A sharp, and the middle note is a B.