The essential difference between the Avatars of Lord Vishnu and Shiva is that while Lord Vishnu is born from the womb as in Ramavatar,Krishnshnavatar,Parashururamavatar and the like, as far as I know Lord Shiva is not recorded of having been born of a womb.
The explanation given is that Lord Shiva is ‘Ayonija’, one who does not stay in the womb.
However there are Avatars of Lord Shiva,Sixty Four in Number are narrated in the Tamil Texts of yore, ‘Thiruvilayadal Puranam’
I have not come across anything other than this in Texts.
In all these instances, Lord Shiva is not born to any one;He appears, attends to the job on hand and disappears.
I have found a list of Avatars of Lord Shiva in Koorma Purana.
The Twenty eight Avatars of Lord Shiva are,
7. Jai Kishhavyan.
I do not fond any reference to these Names in the Vedas.
Some of the names mentioned here Pinda Muneeswaran,Muni are the names of the Grama Devatas or Gods protecting the Towns, who are generally described as belonging to lower Pantheon of Gods in Tamil Nadu.
Source: Koorma Purana.
There is again a mention of Lord Shiva’s Amsa, a reflection of Him in…as,
Sun as Rudra,
Water as Bhava,
On Earth as Siva,
As Pasupathy in Fire,
As Esa in Air,
As Bhima in Heavens,
As Fire in the bodies of Brahmins and as
Mahadeva, in Chandra, Moon.
Normally the Vedas describe Lord Rudra as an Amsa of lord Shiva and here it is the other way!
Again most of the references here are more in the nature of explaining the essentials of the Nature of the elements where Shiva is described to be residing in.
And to top it all, Lord Rudra is described as the Son of Lord Brahma.
Contributions on this subject is welcome.
- What Is Yoga Iswara Gita By Vyasa (ramanan50.wordpress.com)