No one has ever been able to say who is the bigger between the two Shiva or Vishnu. As per various holy texts and Puranas, Shiva and Vishnu the two of the holy trinity of Hinduism share the most special bond of love, devotion, and admiration for each other. There are no stories of differences or fights between the two gods.
1. Shiva’s Sharbha Avatar vs Vishnu’s Narsimha Avatar
However, there are stories in Puranas about the fight between the incarnations of these two avatars. The first story is about the fight between Shiva’s Sharbha avatar and Vishnu's Narasimha avatar, which is found in the Shiva Purana. Lord Vishnu assumed the most ferocious form of Narasimha to kill the asura king Hiraniya Kashyap, the father of Prahalad.
Even after the killing, Hiraniya Kashyap's Narasimha rage did not calm down and his wrath threatened to destroy the world. No one in the universe dared to face lord Narasimha's anger. At the request of the devas Shiva sent, Veerbhadra to tackle Narasimha, but Veera Bhadra could not win over Narasimha when that failed, Shiva manifested as Sharbha Shiva, assumed the form of a part, animal and a part bird named Sharbha to tame Narasimha down. Sharbha's avatar had the body of an eight-legged lion having two heads of a beast: two wings and a long tail. After failing at various polite attempts to calm Narasimha, Sharbha's avatar attacked Narasimha. In the battle against Narasimha, Sharbha tied Narasimha to his tale, and immobilized him, thus eventually calming Narasimha's rage.
2. Shiva’s Vrishabha avatar vs Vishnu’s Immoral Sons
The second story is from the Shatarudra Samhita of the Shiva Purana. That speaks about another event when Vishnu had to fight a battle against Shiva’s Vrishabha avatar. However, the authenticity of the story is questionable, as no similar texts are found in any other Purana. During the Samudra Manthan or the churning of the ocean, Vishnu tricked the asuras by taking the most alluring form of Mohini.
However, the Shiva Puran has a modified version of the story. In which a bunch of beautiful apsaras come from the churning of the ocean, the asuras enchanted by their beauty kidnap the apsaras and take them to their abode the Patal Lok.
After that, they come back to take control of the Amrit, but the devas had already taken control of the Amrit, which resulted in a huge fight between the devas and the asuras. Vishnu to save the devas fought the battle against the asuras and, in the course of fighting he chases them to Patal Lok, after defeating the asuras in Patal Lok, Lord Vishnu comes across the beautiful apsaras and becomes enchanted by their beauty. Struck by the arrow of Kama, he fathered many immoral sons who went on to create a ruckus for the devas.
Those sons of Vishnu had great strength and valor and brought havoc both in heaven and earth, causing misery to all. On seeing the harm done to the world, the devas and the sages approached Brahma and informed him of this. Then Brahma goes to Shiva informing him of the mayhem caused by the sons of Vishnu to get rid of these cruel sons of Vishnu. Lord Shiva assumed the form of an ox named Vrishabha and killed all the brutal sons of Vishnu. Vishnu attacked, Shiva’s Vrishabha avatar, with arrows and missiles but Shiva in the form of a bull stood like a mountain and got hold of all those missiles of Vishnu.
After a brief battle, Vishnu realized that it was Shiva who he was fighting against. He comes out of his Maya and returned to his abode.
3. Vishnu’s Krishna Avatar vs Lord Shiva
The third story is about the fight between Shiva and Vishnu's Krishna avatar. That is a part of Vishnu Parva. This story also forms a part of the Srimad-Bhagavat Purana. Banasura is the 10th descendant of Prahalad and the eldest among the hundred sons of Bali. Since he was fond of lord Shiva from childhood. He meditated and worshiped hard to appease Lord Shiva. Lord Shiva appeared before him and gave him the boon of a thousand powerful arms which would make him very powerful. Banasur asked for the boon that Shiva would always protect him whenever he was in trouble.
Shiva granted him that boon too. Banasur had a daughter named Usha, who was in love with Anirudh the grandson of Krishna upon finding out about the secret love between Usha and Anirudh, Banasura abducted Anirudh and held him captive. So, Krishna came with Balarama and the armies of Dwaraka to defeat Banasura and rescue his grandson. Balarama with his power destroyed all of Banasura's army Banasura confronted Krishna, but he was powerless in front of Krishna. Then Banasura prayed to Lord Shiva for protection.
Lord Shiva was confined with his boon and came to rescue Banasura and started the war with Krishna. To protect Banasura, both Lord Shiva and Lord Krishna battled with each other when Krishna found out that he couldn’t kill Banasura, because lord Shiva was protecting him. He prayed to Lord Shiva in silence, then Shiva gave him a trick to follow. Krishna threw an arrow that made lord Shiva fall asleep after this Krishna took out his Sudarshan chakra and cut off all the arms of Banasura.
Finally, he ordered the Sudarshan chakra to cut off the Banasaur’s head only when Shiva woke up, so that his Sudarshan chakra could be blocked by Shiva, while all the stories we narrated so far were between the incarnations of God.
4. Vishnu’s Sharanga vs Shiva’s Bow
There is only one story that talks about the battle between Shiva and Vishnu themselves. This story is found in the 75th Sarga of Bala Kanda of Ramayana, where Parashuram narrates the story to Lord Ram before challenging him to string Sharanga. Once all the devas asked Brahma as to who was more powerful between the blue-throated Lord Shiva and Sudarshan Chakradhari, lord Vishnu, lord Brahma, tried to create a difference of opinion between the two to find out. Who was the superior one? In the battle that ensued, Lord Vishnu uttered a battle cry “hum”, which paralyzed the devas, the overpowering bow of Shiva was broken and Shiva was frozen. His bow was rendered useless and Lord Vishnu’s bow Sharanga had prevailed. From then on, the gods, along with the assemblages of sages deemed Vishnu to be paramount.