After Pegasus was born he was tamed by Athena and given by her to the Muses on Mount Helicon. There were also water nymphs of springs and wells called Pegasides that were connected to Pegasus.[iii]
In Greek mythology horses were often associated with water due to their connection with Poseidon. Poseidon is not only the god of the sea, but also of horses. Pegasus is also linked with water because, like his father, of his ability to create springs, only his were created with a strike of his hoof. The most famous of these was Hippocrene on Mount Helicon, the home of the Muses. It was believed that the waters of this spring inspired creativity, in particular for poets.[iv]
With Athena’s help, Pegasus was captured at the Corinthian well of Pirene by the hero, Bellerophon, who he helped defeat the Chimaera and others. [v] [vi] Later Bellerophon became prideful and decided to ride Pegasus to Mount Olympus. Zeus in his anger sent an insect to bite Pegasus, causing him to throw Bellerophon to his demise. [vii]
Pegasus then flew on to Olympus, with Athena, where he joined the other horses of Zeus, and was given the task of carrying Zeus’ thunder and lightning. To honor his service, Zeus made Pegasus into a constellation.
The 9 daughters of Zeus & Mnemosyn (Titaness goddess of memory)
Calliope / Kalliopē [meaning beautiful-voiced] – The muse of epic poetry, symbolized by the writing tablet.
Clio / Kleio [meaning to make famous] – The muse of history, symbolized by the scroll or book.
Erato [meaning desired] – The muse of love poetry, symbolized by a lyre and crown of roses.
Euterpe [rejoicing well] – The muse of music, symbolized by the double flute.
Melpoménē [meaning to sing] – The muse of tragedy, symbolized by the tragic mask or sword.
Polyhymnia/Polymnia [meaning many hymns] – The muse of sacred poetry, also of agriculture, symbolized by grapes & a veil.
Terpsichore [meaning delight in dancing] – The muse of dance, symbolized by the lyre.
Thalia [meaning to be joyous & flourish, & to be verdant] – The muse of comedy, symbolized by comic mask and the shepherd’s hook.
Urania [heavenly] – The muse of astronomy & philosophy, symbolized by a globe and compass.
“HELICON, a mountain range, of Boeotia in ancient Greece, celebrated in classical literature as the favourite haunt of the Muses, is situated between Lake Copais and the Gulf of Corinth. On the fertile eastern slopes stood a temple and grove sacred to the Muses, and adorned with beautiful statues, which, taken by Constantine the Great to beautify his new city, were consumed there by a fire in A.D. 404. Hard by were the famous fountains, Aganippe and Hippocrene, the latter fabled to have gushed from the earth at the tread of the winged horse Pegasus, whose favourite browsing place was there. At the neighbouring Ascra dwelt the poet Hesiod, a fact which probably enhanced the poetic fame of the region. Pausanias, who describes Helicon in his ninth book, asserts that it was the most fertile mountain in Greece, and that neither poisonous plant nor serpent was to be found on it, while many of its herbs possessed a miraculous healing virtue. The highest summit, the present Palaeovouni (old hill), rises to the height of about 5000 ft. Modern travellers, aided by ancient remains and inscriptions, and guided by the local descriptions of Pausanias, have succeeded in identifying many of the ancient classical spots, and the French excavators have discovered the temple of the Muses and a theatre. ” – 1911 Encyclopædia Britannica
[i] So Perseus stood over them as they slept, and while Athena guided his hand and he looked with averted gaze on a brazen shield, in which he beheld the image of the Gorgon,5 he beheaded her. When her [Medusa’s] head was cut off, there sprang from the Gorgon the winged horse Pegasus and Chrysaor, the father of Geryon; these she had by Poseidon. – Apollodorus, Library http://www.perseus.tufts.edu/hopper/text?doc=apollod.%184.108.40.206&lang=original
[ii] “ And when Perseus cut off her [Medusa’s] head, there sprang forth great Chrysaor and the horse Pegasus who is so called because he was born near the pegae (spring or well) of Ocean[us]; and that other, because he held a golden blade in his hands. Now Pegasus flew away and left the earth, the mother of flocks,  and came to the deathless gods: and he dwells in the house of Zeus and brings to wise Zeus the thunder and lightning.” – Hesiod, Theogony (8th-9th century) http://www.perseus.tufts.edu/hopper/text?doc=Hes.+Th.+270&fromdoc=Perseus%3Atext%3A1999.01.0130#note-link2
[iii] “ Pegasides; the Muses, a word of Greek derivation, from πήγη, a spring; for Pegasus is said to have opened the Helicon fountain by a stroke of his hoof.” – Commentary on the Heroides of Ovid http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0061:poem=15&highlight=pegasus
[iv] “When Euripides says, “‘I come from the Acrocorinthus, well-watered on all sides, the sacred hill and habitation of Venus,’” the epithet ‘well-watered on all sides,’ must be understood to refer to depth; pure springs and under-ground rills are dispersed through the mountain; or we must suppose, that, anciently, the Peirene overflowed, and irrigated the mountain. There, it is said, Pegasus was taken by Bellerophon, while drinking; this was a winged horse, which sprung from the neck of Medusa when the head of the Gorgon was severed from the body. This was the horse, it is said, which caused the Hippocrene, or Horse’s Fountain, to spring up in Helicon by striking the rock with its hoof.” – Strabo, Geography (Book 8 CH 6) http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.01.0239:book=8:chapter=6&highlight=pegasus
[v] “… Bellerophon, who once suffered greatly when beside the spring he wanted to harness Pegasus, the son of the snake-entwined Gorgon;  until the maiden Pallas brought to him a bridle with golden cheek-pieces. The dream suddenly became waking reality, and she spoke: “Are you sleeping, king, son of Aeolus? Come, take this charm for the horse; and, sacrificing a white bull, show it to your ancestor, Poseidon the Horse-Tamer.”  The goddess of the dark aegis seemed to say such words to him as he slumbered in the darkness, and he leapt straight up to his feet. He seized the marvellous thing that lay beside him, and gladly went to the seer of the land,  and he told the son of Coeranus the whole story: how, at the seer’s bidding, he had gone to sleep for the night on the altar of the goddess, and how the daughter herself of Zeus whose spear is the thunderbolt had given him the spirit-subduing gold. The seer told him to obey the dream with all speed;  and, when he sacrificed a strong-footed bull to the widely powerful holder of the earth, straightaway to dedicate an altar to Athena, goddess of horses. The power of the gods accomplishes as a light achievement what is contrary to oaths and expectations. And so mighty Bellerophon eagerly  stretched the gentle charmed bridle around its jaws and caught the winged horse. Mounted on its back and armored in bronze, at once he began to play with weapons. And with Pegasus, from the chilly bosom of the lonely air, he once attacked the Amazons, the female army of archers,  and he killed the fire-breathing Chimaera, and the Solymi. I shall pass over his death in silence; but Pegasus has found his shelter in the ancient stables of Zeus in Olympus.” – Pindar,Olympian Ode (8th century) http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0162%3Abook%3DO.%3Apoem%3D13
[vi] “ So Bellerophon mounted his winged steed Pegasus, offspring of Medusa and Poseidon, and soaring on high shot down the Chimera from the height. After that contest Iobates ordered him to fight the Solymi, and when he had finished that task also, he commanded him to combat the Amazons. And when he had killed them also, he picked out the reputed bravest of the Lycians and bade them lay an ambush and slay him. But when Bellerophon had killed them also to a man, Iobates, in admiration of his prowess, showed him the letter and begged him to stay with him; moreover he gave him his daughter Philonoe, and dying bequeathed to him the kingdom.” Apollodorus, Library http://www.perseus.tufts.edu/hopper/text?doc=Hes.+Th.+270&fromdoc=Perseus%3Atext%3A1999.01.0130#note-link2 “If a man looks to things far away, he is too short to reach the bronze-floored home of the gods; winged Pegasus threw his master Bellerophon, who wanted to go to the dwelling-places of heaven and the company of Zeus. A thing that is sweet beyond measure is awaited by a most bitter end.” – Pindar, Isthmean, 7:40 (8th century) http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.01.0162:book=I.:poem=7&highlight=pegasus