In many cultures, there was a god, goddess or muse for dance.

goddesofthedanceThe bhakti sects of Shaivism (Hinduism) see Shiva as the one true God. He is often depicted as a dancer of the tantric Tandava dance.
The Natarajasana is a pose, a dancing figure of Shiva and is also used as a symbol (in yoga). This is when one foot with one or two hands is held up behind the back.
The erotic dance done by his wife Goddess Parvati is known as Lasya.

The Japanese goddess' Uzume’ ('whirling') was named Daughter of Heaven and is the goddess of joy and dance.

Bastet (originally Bast, whether also Ubasti, Bubastet) was in the Egyptian mythology the goddess of fertility, which was presented in the form of a cat. In early versions with a lion head and an ankh cross. It is the goddess of joy, dance, music and celebration.
Hathor / Meh-weret was a goddess with many functions: ruler of the heavens, living soul of trees, wet nurse of the kings, goddess of love and dance.

The Greek Hermes, son of Zeus, and messenger who was often depicted with a lyre and winged feet, had as Roman equivalent Mercury. They are the god or guardian of the dancers.
Hermes' son Pan, the god of the forest, usually with goats feet, is sometimes called Lord of the Dance or dance master.

Dionysos or Dionysus was the god of agriculture, fertility, nature, wine, fun and dance, life, and immortality.

A satyr is a (mythological) cheerful and mischievous forest creature, companion of the god Dionysus. They have a bucks tail, -ears and sometimes -legs, and ithyphallos (erected penis) and lots of hair and a beard. They often carry a whistle. Known satyrs are Marsyas and Aristaeus. (The god Pan is very similar to a satyr.)
The satyr loves wine and tempting nymphs (graceful demigods from nature) and ephebes (boys). They stand for lust and ecstasy (just like the female followers of Dionysus, the Maenads or Bacchae).
Satyrs let instinct go above reason, anarchy above order, ecstasy above asceticism, abundance above moderation.

Dionysos was known to the Romans as Bacchus.

The muses were representations of a goddess as an inspiring force for people.
The nine muses (or Parnassides) were the Greek goddesses of the arts and science, daughters of Zeus and the sisters of Apollon. They mostly stayed on a mountain as the Parnassus or the song mountain Helikon.

Terpsichore is the muse of dance and lyric poetry as an attribute with the lyre.
Euterpe is the muse of the flute
Erato is the muse of the hymn, the song and lyric
Kalliope is the muse of heroic poetry, philosophy and rhetoric
Clio is the muse of history
Melpomene is the muse of tragedy
Polyhymnia is the muse of rhetoric and sacred songs
Thalia is the muse of comedy
Urania was the muse of astronomy

The Romans identified the Muses later with Italian source nymphs, the Camenae.

In the Catholic world there is also a saint for the dance (Saint Vitus, see patron saint), and a Lord of the dance. (See two additional articles.)