Cult
The Muses had several temples and shrines in ancient Greece, their two main cult centres being Mount Helikon in Boiotia and Pieria in Makedonia. Strabo wrote: 

"Helikon, not far distant from Parnassos, rivals it both in height and in circuit; for both are rocky and covered with snow, and their circuit comprises no large extent of territory. Here are the temple of the Mousai and Hippukrene and the cave of the Nymphai called the Leibethrides; and from this fact one might infer that those who consecrated Helikon to the Mousai were Thrakians, the same who dedicated Pieris and Leibethron and Pimpleia [in Pieria] to the same goddesses. The Thrakians used to be called Pieres, but, now that they have disappeared, the Makedonians hold these places. "The cult of the Muses was also commonly connected to that of Apollo. Emblems
Some Greek writers give the names of the nine Muses as Kallichore, Helike, Eunike, Thelxinoë, Terpsichore, Euterpe, Eukelade, Dia, and Enope.In Renaissance and Neoclassical art, the dissemination of emblem books such as Cesare Ripas Iconologia (1593 and many further editions) helped standardize the depiction of the Muses in sculpture and painting, so they could be distinguished by certain props. These props, or emblems, became readily identifiable by the viewer, enabling one immediately to recognize the Muse and the art with which she had become associated.
According to a myth from Ovids Metamorphoses—alluding to the connection of Pieria with the Muses—Pierus, king of Macedon, had nine daughters he named after the nine Muses, believing that their skills were a great match to the Muses. He thus challenged the Muses to a match, resulting in his daughters, the Pierides, being turned into chattering jays (with κίσσα often erroneously translated as magpies) for their presumption.Pausanias records a tradition of two generations of Muses; the first are the daughters of Ouranos and Gaia, the second of Zeus and Mnemosyne. Another, rarer genealogy is that they are daughters of Harmonia (the daughter of Aphrodite and Ares), which contradicts the myth in which they were dancing at the wedding of Harmonia and Cadmus. Children
Calliope had two sons, Ialemus and Orpheus, with Apollo. In another version of the story, the father of Orpheus was Oeagrus, but Apollo adopted him and taught him the skill of lyre while Calliope trained him in singing. Linus was said to have been the son of Apollo and one of the Muses, either Calliope or Terpsichore or Urania. Rhesus was the son of Strymon and Calliope or Euterpe. The sirens were the children of Achelous and Melpomene or Terpsichore. Kleopheme was the daughter of Erato and Malos. Hyacinth was the son of Clio, according to an unpopular account. Hymenaeus was assigned as Apollos son by one of the muses, either Calliope, or Clio, or Terpsichore, or Urania. Corybantes were the children of Thalia and Apollo.