Let me preface this by saying that since most of our scriptures were composed or compiled by Vyasa at the end of the Dwapara Yuga or the beginning of the Kali Yuga, most descriptions of Shiva's activities in the Kali Yuga would be in the form of prophecies. But here are at least some potential activities that Shiva may have done in the Kali Yuga:

  1. One thing we know for sure that Shiva did in the Kali Yuga was attend the wedding of Venkateshwara (the Vishnu deity in Tirupati, aka Balaji or Srinivasa. Here is what this excerpt from the Venkatachala Mahatmya of the Skanda Purana says:

    Then [Venkateshwara] proceeded towards Narayanapuri accompanied by Brahma, Isha [Shiva], Varuna, Yama, Yakshesha, Vasishta and other eminent sages, Sanaka and other Yogins as well as devotees and loyal followers of the Lord. The chieftains of Gandharvas sang. The groups of celestial damsels danced.

    And when Vishnu took a loan from Kubera to pay for the wedding expenses, Shiva was one of the witnesses, as described in this excerpt from the Sthala Purana (temple scripture) of Tirupati (which you can read here):

    "The entire Universe is at your beck and call. All my money is yours. I have no control over it. You are its master" said Kubera [to Vishnu]. Finally, Kubera agreed to lend the money required if the necessary document was executed by the debtor. Brahma then dictated a promissory note. The note was executed.

    "This, the 7th day of the bright-fortnight of the month of Vysakha in Kaliyuga, The debtor is Srinivasa, the creditor is Dhaneswara. Purpose: marriage of the loanee. Amount of loan fourteen lakhs in Ramamudra coins. The loanee should repay it with interest in one thousand years after the loanee's marriage." Brahma and Siva attested as witnesses. The scribe was the loanee himself. After accepting the promissory note, Kubera paid the amount in cash.

    The loan was handled by another incarnation of Vishnu named Govindaraja, as I discuss here. By the way, The "one thousand years" means years of the gods, which I discuss here; it basically means that the period of the loan would be the entire Kali Yuga. That is why the Tirumala Venkateshwara Temple in Tirupati created the custom of having an Hundi, a box where people could donate money in order to pay off the interest on Kubera's loan. Of course, other temples realized that it was a valuable way to fund temple operations, and nowadays Hundia are a standard feature in Hindu temples.

  2. Many people believe that Adi Shankaracharya, the famous proponent of Advaita Vedanta, was an incarnation of Shiva. Some people dispute this because of the story of Shiva appearing before Adi Shankaracharya as a Chandala. But in any case, there are three quotes from Hindu scripture that various people cite as prophecies of Adi Shankaracharya. The first is from this excerpt from the Kurma Purana, which describes the Kali Yuga:

    Shankara, Nilalohita, will take up incarnations for the purpose of establishing the Shrauta (Vedic) and Smarta (belonging to the Smritis) rites, with a desire for the welfare of his devotees. He will teach his disiciples the knowledge pertaining to the Brahman.

    And then there are two quotes from the Padma Purana, which if genuine and if correctly identified with Adi Shankaracharya, would paint Advaita Vedanta in a negative light. The first is from this excerpt from the Uttara Kanda of the Padma Purana, where Vishnu says this to Shiva:

    Being born in the ages like Dvapara and among men in the Kali Yuga make the people averse to me with the sacred texts prepared by you. So also censure me so that the world become better and better. I shall produce this delusion which will delude people. You too, O mighty-armed Rudra, produce sacred texts that would delude people. O you of great arms, produce (the texts) that are false and spurious. Manifest thyself and keep me hidden.

    Genuine or not, I think it's possible that this verse is referring to Shiva's incarnation Lakukisha, whom I discuss below, rather than Adi Shankaracharya. But in any case, by far the most famous quote that people claim to be a prophecy of Adi Shankaracharya is from this excerpt from the Uttara Kanda of the Padma Purana, where Shiva says this:

    Vishnu of the form of Buddha proclaimed the false Buddhist doctrine and those of the naked and wearing dark blue garments for the destruction of demons. The doctrine of Maya (illusion) is a wicked doctrine and said to be pseudo-Buddhist. I myself, of the form of a brAhmana, proclaimed it in Kali (age).

    It shows the meaninglessness of the words of the holy texts and is condemned in the world. In this (doctrine) only the giving up of one's own duties is expounded. And that is said to be religiousness by those who have fallen from all duties. I have propounded the identity of the Highest Lord and the (individual) soul. I stated this Brahman's nature to be qualityless. O goddess, I myself have conceived, for the destruction of the worlds, and for deluding the world in this Kali age, the great doctrine resembling the purport of the Vedas, (but) non-Vedic due to the principle of Maya (illusion) (present in it).

    Of course, Advaitins would dispute the authenticity of these verses, and I'm inclined to agree with them. Still, there's an interesting analytical point here, which is that Advaita bears some similarity to Buddhist thought. In fact, some scholars believe that Adi Shankaracharya's guru's guru Gaudapada was originally a Yogachara Buddhist, although as I discuss here Advaita tradition identifies Gaudapada's guru as Vyasa's son Shuka. In any case, for more information on how Vaishnavism views Advaita, see my answer here.

  3. In this excerpt from the Shatarudra Samhita of the Shiva Purana, Shiva tells Brahma about his various Yogeshwara incarnations, which he takes once every Mahayuga. Here is what he says about the present Mahayuga:

    In the twenty-eighth aeon of Dvapara, there will be Dvaipayana Vyasa, the son of Parashara ,and the most excellent of Purushas [Vishnu] shall be born as Krishna with his one-sixth part, as the foremost of the sons of Vasudeva. Then I too shall be born with the body of a Brahmachari and the soul of a Yogi by means of Yogic Maya to the great surprise of the worlds. On seeing a dead body forsaken in the cremation ground I shall enter into it and make it free from ailments by means of Yogic Maya for the welfare or Brahmins. Then I will enter the holy divine cavern of Meru along with you and Vishnu. O Brahma, I shall then be known as Lakulin.

    Lakulin, also known as Lakulisha, was a famous Shaivite thinker who introduced Pashupata, an ancient sect of Shaivism which among other things involves smearing ash all over your body; Shiva is described as the originator of the Pashupati vow in this chapter and this chapter of the Shanti Parva of the Mahabharata.

    But it's not clear to me what Lakulisha's life story is, or whether he lived during the Dwapara Yuga or the Kali Yuga, that is why I asked this question.

Let me just add that apart from these three, there are probably other stories involving Shiva in the Kali Yuga. Just as the Alwars, the 12 Vaishnava poet-saints I discuss in my questions here, are said to have encountered Vishnu on various occasions, I expect that if you looked at the life stories of the 63 Nayanars, Shaivite poets living at about the same time, you may find stories of various run-ins with Shiva. But I'm not sure if Hindu scripture contains any prophecies of the Nayanars.

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