Vishnu

Vishnu datta is a person who had studied sastras and does all swakarmas in the manner instructed in the sastras. He does Vaiswadevam daily(a karma as per Vedas) and pooja to the guest. While Vishnu datta was waiting for the guest as usual he saw an ugly looking beggar at the door. As per Vedas, one who comes for food is Vishnu. Vishnu data took the food and gave it to the beggar. Seeing the shining in the eyes, Vishudatta had a feeling that this was not a beggar. The beggar went out running and Vishnu datta followed him. The beggar beat Vishnu datta with stones and branches of trees. Vishnu datta however managed to catch hold of the feet of the beggar. Lord Dattathreya in the form of the beggar gave mantropadesa and blessed Vishnu datta.

VIishnu datta  was tested thrice in the form of spirits and as he became victorious he was given a lot of boons and mantropadesas making him elite. Vishnu datta made a lot of people redress their grievences with the powers received from Dattathreya. With the help of Datta mantras, a Brahman boy with hot temper and bad manners was corrected. When somebody does his dharma, Bhagavan gives darsan and moksha.

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According to Mahanubhava dharm, lord Dattatreya is like Sri Krishna is god himself single headed and not triple headed. They believe he is not an incarnation of the three devas, Brahma, Vishnu and Mahadev, but is the supreme lord himself.

http://en.wikipedia.org/wiki/Mahanubhava
http://mahanubhavdarshan.blogspot.com/
https://www.youtube.com/results?search_query=Mahanubhav
http://www.facebook.com/mahanubhav.panth

https://www.google.com/search?q=Mahanubhava+incarnations

Soorya

There are three holy lakes in Dattathreya ashram known as

1. Mathru theertham, 
2. Meru theertham and 
3. Padma theertham. 

The sanctity of the lakes increased in the presence of Lord Dattathreya and even Bhagavan Soorya was relieved of his grave mistakes and paapa. He also got relief from diseases by taking a dip in these lakes. Lord Soorya returned to his place purified and healthy by the blessings of Dattathreya.

Amalakasramam

When the Devas and Asuras churned palazhi with Mandara mountain and Vasuki as the rope, Godess Lakshmi along with a goosberry tree(Malayalam- nelli, Sanskrit- amalaki, Botanical name- Philanthus emblica) came out of it . Lord Mahavishnu accepted Godess Lakshmi and took the tree to Vaikunda. When Brahama Deva approached Lord Dattatraya as he forgot the Vedas, he planted a twig of this tree brought from Vaikunda at Sahya hills. Parasurama worshipped Lord Maheswara under this tree and hence gooseberry tree got the blessings of Lord Siva. So this tree became dear to not only trinities but also Lord Datatreya. Since Amalaki tree was growing in Dattathreya ashram it was called Amalakasramam.

Brihaspati

When the demon Jambasura saw that his friend Bali had fallen, he appeared before Indra, the enemy, just to serve Bali Maharaja with friendly behavior. The greatly powerful Jambasura, carried by a lion, approached Indra and forcefully struck him on the shoulder with his club. He also struck Indra's elephant. (Srimad-Bhagavatam 8.11.13-14)

Devendra(king of devas) fought with him for a lot of years. Devas became tired in the war and Devendra approached devaguru(the teacher of devas), Brahaspathy for advice. 

Brahaspathy told him that even the trinities(Brahama,Vishnu,and Mahesa) could not save them and the only way is to approach Sri Dattathreya the incarnation of trinities. Devaguru also told Indra that Dattathreya always tests his devotees and strong belief and prayers could only please him. As advised by Brahaspathy, Indra went to Sahya hills to meet and obtain blessings of Dattathreya. Reaching there, he found that Swami is drinking in the middle of his followers performing dances after drinks. Devendra without any hesitation started to continue the services performed by his devotees and disciples. Dattathreya asked him why he is performing all these mean deeds like a servant following him and Indra said you are an incarnation unaffected by any mean deeds or practices, more over you are the greatest guru. Indra continued to serve Dattathrya like an earnest servant and honest disciple. One day Swami asked Indra why he is serving him the most incapable. Indra revealed the purpose of his visit . Dattathreya offered help to kill Jambasura. Indra fought Jambasura as advised by Narada maharshi, withdrew from the war and made Jambasura reach the Sahya hills. Jambasura found Anakha Devi in the shape of an extremely beautiful lady and proposed to marry her. Devi replied that if Jambasura shows the ability to take her on to his head, she will marry him. In the meantime, Lord Dattatreya weakened the strength of Jambasura. Jambasura however managed to lift the Devi to his head and he became ashes and Indra became victorious. 

https://trinitygods.blogspot.com/search/label/Jambhasura

Body

Anasooya Devi, his mother was happy to see her son Dattatherya returning to the ashram. Dattathreya requsted the permission of his parents to go over to Sahya mountains for the welfare of the devotees. Even when Anasooya Devi was familiar with continuous meditation, at first she did not grant permission to her son due her fondness for her child. Dattathreya reminded his mother of the greatness of meditation and finally obtained permission. Anasooya devi said that she had claim on his physical body and asked him to refuse it. Dattathreya tore his body and submitted it at the feet of his mother. The mother realized that meditation is important to him as he is dear to her and she as a mother should be broad hearted, and sacrifice her pleasures for the benefit of the world. The regreted Anasooya Devi gave back the physical body of Dattathreya and granted permission, and blessed him with tears. Dattathreya started meditation at the Sahya hills with the permision of his parents. He not only uplifted human race but also Indras and other devadas. Thus the acts of Lord Dattathreya beyond common man's imagination was for the welfare and wellbeing of the entire world.

https://www.youtube.com/results?search_query=Sahya+hills

acting

In order to test the eligibility of the people of the ashram who asked him to accept them as his devotee, Dattathrya plunged in to the waters of a lake and disappeared. The devotees waited around the lake for hundred years meditating in order to get wonderful powers.

At the end of the waiting for hundred years, Swami appeared over the water with a bowl full of liquor in one hand and a beautiful lady on the other. in order to test the devotees, Swami was chatting, dancing, flirting, singing, drinking, and smoking with the girl. The rishis who saw this thought that Swami has deviated from morals and left him. Some rishis remaining said that some have left without understanding that he was actually acting and they requested him to guide them. Hearing their request, Swami took them all to the ashram as followers and disciples(sishyas).

Sahyadri

Sree Dattathreya stayed in the caves of Sahyadri as yogiswara(the lord of the yogis), a guide to moral and religious people, a kalpataru(tree which provides every thing demanded) to devotees and savior of the universe.

https://en.wikipedia.org/wiki/Western_Ghats
http://www.sahyadrischool.org/
http://www.westernghat.org/
https://sahyadri.edu.in/

Atrishwara

Atrishwara Linga in a forest named Kamda on Chitrakuta Mountain where the Maharshi saved people from prolonged drought for some fifty years; once Arti fainted out of thirst and his wife Anasuya went in search of water; Devi Ganga appeared and offered water from a spring and Anasuya carried a Kamandalu-full of water and quenched Atri’s thirst. It was from that Spring emerged the Sacred Mandakini River and as the result of the Tapasya, the prolonged drought ended and Bhagavan Shiva appeared as the Atrishwara Linga.

https://shivamahapurana.wordpress.com/2012/02/26/atrishwara-incarnation-greatness-description/

Pose

1. Vishna Master of Sleep Pose
2. Shiva Is Master of Sit Pose
3. Brahma Is Master of Stand Pose