Foremost, God Shiva didn’t take any avatars. Shiva means the untouched one(who doesn’t take any birth and so there is no death). He is the ultimate consciousness.
There are many forms of God Shiva. Here forms mean, God Shiva according to the dire need of the Universes and its resident’s well-being, manifests Himself in many form and upholds dharma. So, Manifestation here doesn’t mean an Avatar. The following are examples for God Shiva’s Ansh(child/form/ possessing few qualities).
- Rudra, it’s the most ferocious form of God Shiva. Rudra word means the roaring sound. He is represented in the form of fire. In this form, He annihilates the evil forces in the multiverses and destroys our ego and gives us salvation from this illusion called world.
- Veerabhadra : When Sati Devi immolated herself, Shiva angrily pulled one of His hair strand and Veerabhadra took birth from that hair strand. He destroyed the Daksha Yagna and beheads Daksha Prajapathi.
- Virupaksha, the fiercely wide eyed God. Mahadev manifested in this form when Sati Devi immolated herself when subjected to humiliation at her father’s abode. God Shiva not able to bear the pain of losing her beloved wife, transformed into Virupaksha and destroyed all with His immense anger.
- Kalabhairava, this form was taken by God Shiva to behead Lord Brahma’s head, to relieve him from his arrogance and Kalabhairavas are the protectors of all Shakthi Peetas(the places where Sati Devi’s body parts fell on Earth after her immolation).
- Dakshinamoorthy, this form of Mahadev is the most pleasant one. In this form, Mahadev sits in the south direction under a banyan tree and preaches to the sapta Rishis(the seven sages) as the Adi Yogi and through only silence.
- Kapali, this form of God Shiva resides in the Rudra Kshetra(cremation grounds) and wears the munda mala(garland of skulls) and in His hand holds the kapal(Lord Brahma’s skull in His hand). Whenever Goddess Parvathi takes an avatar and at the end of her every avatar, one skull is added to the garland of skulls.
- Pashupatinath is a prominent form of God Shiva meaning the "God to all living beings and animals”. This form signifies Shiva as the master who frees all the human souls, including animals from worldly illusion and ignorance (pashamu or attachments). In this form God Shiva has the most powerful weapon called Pashupatastra.
- Hanuman(the eleventh Rudra), Mahadev gave His powers to Hanuman in order to help Shri Rama upload dharma in the Tretayuga and thus kill King Ravana.
- Nataraja, in this form, God Shiva does the cosmic dance and in the end of this Kaliyuga, everything will get dissolved into Shiva and this world will end.
- Ardhanareeswarar, as the name suggests it’s half woman and half man. This portrays that male and feminine energies are equal and the world will cease to exist if there is misbalance among them. God Shiva gave His half body to His better half Goddess Parvathi and thus came the concept of Ardhanareeswarara. Mahadev took this form to explain Sage Bhringi that men and women are equal.
- Sharabha is a fierce, powerful manifestation of God Shiva combining the features of a lion and a bird, it was to pacify Lord Narasimha. According to the Shiva Purana, this form features eight legs, a lion-like face, and wings, which portrays the restoration of cosmic balance.
Hara Hara Mahadeva Shambho Shankara