This day is celebrated as Kurma Jayanti. The Kurma Puran authored by Ved Vyas, details narrations of the various pastimes of this incarnation of the Lord.
Various other scriptures such as the Ramayan and Mahabharat too detail the episode of Samudra Manthan - Churning of the ocean.
In the eighteenth chapter of Vasudev Mahatmya the Lord says, 'When the sons of Kashyapa churn the ocean with the Mandara Mountain as the churning rod, which has no base to rest on, I will carry that Mandara on my back, assuming
the form of a great tortoise'. (Verse 19)
In this incarnation, the Lord Vishnu manifested as a tortoise. Due to Sage Durvasa's curse, the demi-gods lost all their wealth. The sage out of respect presented a garland of flowers to Indra who carelessly ignored his humbleness. Indra's elephant trampeled the garland, hence the sage in rage cursed Indra to loose all his wealth in to ocean. You can read more in detail about this legend within Shree Vasudev Mahatmya - English, Gujarati
The Devtas approached Vishnu for help. Vishnu then asked them to churn the ocean of milk after adding medicines into the ocean. Mt Mandara could be used a the churning stick he said. He requested them to ask them help of Asuras in lifting the mountain in exchange for offer of the share of nectar of immortality that would ensue from the churning. Both the devatas and the asuras churned the ocean using the serpent Vasuki as the rope. At the start, playing a Machiavellian trick, Indra, king of the gods asked the asuras for the head end of vasuki. But asuras suspecting foul play, took the head end, only to be deceived as the poison from Vasuki was slowly weakening them. But as churning was proceeding the mountain was sinking and then Lord Vishnu took the form of the tortoise KURMA and kept the mountain afloat. As soon as the bowl of amrita, the nectar of immortality was out, the asuras grabbed it. Then Lord Vishnu took the form of an apsara, a beautiful maiden, and seduced the asuras into letting her distribute the nectar and also to abide by her order of distribution. As soon as the devatas were served the maiden disappeared thus totally deceiving the asuras and making them totally weak.
Our presiding deity (Ishtdev) Lord Shree Swaminarayan performed various pastimes/divine actions (Leelas) on this Earth through His various Incarnations (Avatars). The divine pastimes of these incarnations can be found in various Holy Scriptures.
In the 3rd Vachanamrut of Gadhda Pratham Prakan the Lord says, ‘“Even a person who can constantly see the form of the Lord in his heart, should recall the pastimes performed by Him in His different incarnations in different places.’
This article highlights mentions of this Avatar in the following holy texts.
1. Shree Vasudeva Mahatmya from Skand Puran, Vishnu Khand
2. Shreemad Bhagwat - Canto 8 Chapter 24 - The Lord's Incarnation as a Fish
3. Shree Vishnu-sahasranam from Shanti Parva of Maharahabharat
4. Shreemad Satsangi Jeevan
5. Shree Vachanamrut
6. Shree Harileelamrut Kalash
7. Premanand Kavya
8. Nishkulanand Kavya
9. Shree Thakorji ni Seva riiti
When your day ends and night comes, I will bear the earth full of herbs, Manus and others throughout the night like a ship, in the form of a great fish.
Ved Vyas – Skand Puran, Vishnu Khand, Vasudeva Mahatmya, Chapter 18, Line 18
Suta Purani narrates the various incarnations of the Lord to the revered Sages.
When there was a complete deluge after the period of the Cākṣuṣa Manu and the whole world was submerged within water, the Lord assumed the form of a fish and protected Vaivasvata Manu, keeping him up on a boat. ||15||
Ved Vyas - Shreemad Bhagwata, Canto 1, Chapter 3, Line 15
Grandsire Bhishma (Devvrat) narrates the thousand names of Lord Vishnu to Yudhisthir after the end of the epic war of Mahabharat. The Matsya Avatar is referred to thrice in this divine hymn.
Rohitah - “The Fish” - : One who assumed the form of a kind of fish called Rohita.
Mahaasringah –“The Great-Horned” –: The Lord tied the ship to his great horn and sported in the waters of the deluge during His incarnation as a fish.
Name No. 797/1000
Sringee - “The Horned One” - : One who at the time of Pralaya (cosmic dissolution) assumed the form of a fish having a prominent antenna.
Ved Vyas - Mahaharata, Shanti Parva, Shree Vishnu Sahasranama
Lord Swaminarayan enlightens Acharyas Ayodhyaprasadji and Raghuvirji festivals celebrated in the Chaitra Month with one of them being Matsya Avatar Janmotsav.
Lord Swaminarayan - Shreemad Satsangi Jeevan, Canto (Prakran) 4, Chapter (Adhyay) 60
In the divine nectar of discussions, Lord Swaminarayan Himself enlightens His devotees of the worship of Matsya Avatar.
“To the north of Meru, there are three continents: Ramyak-Khand – where Matsya is worshipped, and Sãvarni Manu is the main Devotee; …”
Lord Swaminarayan – Vachanamrut, Bhugol Khagol
On Wednesday Kartik Sud 12, 1881, Lord Swaminarayan established various forms of Himself in the Vadtal Temple. One of them was Matsya Avatar. He then with various Devotees offered prayers to these forms and also performed Aarati to all of them at the same time.
The Prayer to Matsya Dev:
You manifested as a fish so strong and annihilated wicked Shankhasur;
You saved the Vedas and gained fame;
Seeing such prowess of Yours, Devotees offer their worship unto You;
Please be seated here (In Vadtal), the abode of righteousness ||47||
Acharya Viharilalji Maharaj – Harileelamrut, Kalash 8, Vishram 28, Line 47
Revered Sadguru Nishkulanand Swami describes the pastimes of Matsya Avatar amongst various others in the Avatar Chintamani.
He who is known as Murari (Bearer of the Flute), incarnated as a fish (Matsy-avatar) || Thus protected the Vedas and annihilated the demon Shankhasur ||30||
Nishkulanand Swami - Nishkulanand Kavya, Avatar Chintamani, Line 30
Service to the Lord Shree Vithalnathji, son of Shree Vallabhacharya prescribes the services to the Lord on this auspicious day.
Vithalnathji - Shree Thakorji ni Seva Riti - Page 74