As per chapter 22 (The Lunar Dynasty) of Kurma Purana, Kartavirya Arjuna's sons also had similar question and they approached Saptarshis to get an answer.
This is what Vashishtha and other sages told them:
One worships the god that one chooses. There are no rules in this regard. All gods yield the desired fruit, if properly worshiped.
Further Vashishtha and other sages said:
To the extent that there are rules, there are as follows:
Kings worship Vishnu, Ishvara (Shiva) and Purandra (Indra);
Brahmanas worship Agni, Aditya (Surya), Brahma and Pinakdhari (Shiva);
Devas worship Vishnu;
Danavas worship Trishuldhari (Shiva);
Yakshas & Gandharvas worship Soma (Chandra, the Moon God);
Vidhyadharas worship Vagdevi (Saraswati);
Sidhdhas worship Hari (Vishnu);
Rakshasas worship Rudra Shankra (Shiva);
Kinnaras worship Parvati;
Rishis worship Brahma and Trishuldhari Mahadeva (Shiva);
Women worship Devi, Vishnu, Isha (Shiva) and Bhaskara (Surya, the Sun God);
Grihsthas worship all;
Brahmacharis worship Brahman;
Vanprasthis worship Arka (Surya, the Sun God);
Sanyasis worship Maheshwara (Shiva);
Bhutas worship Bhagvan Rudra (Shiva);
Kushmandas worship Vinayaka;
Prajapati Brahma is god for all;
But Men should worship Hari (Vishnu) after realizing that Rudra (Shiva) is no different from Vishnu. Else, Hari (Vishnu) will not destroy the enemies of the king.
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