What is the ruling on the dancing, singing, and swaying of
during what they call circles of Dhikr (Remembrance of Allah)?
The best of speech is the Book of Allah and the best of guidance is the guidance of Muhammad (peace be upon him). The most evil of things are those which have been newly invented in religion. Allah has completed the religion (Islam) for His servants in all aspects: actions, words and beliefs. He (Exalted be He) says:
This day, I have perfected your religion for you, completed My Favour upon you, and have chosen for you Islâm as your religion.
The Messenger of Allah (peace be upon him) clarified the matters of this religion through his acts, sayings, and approvals. Moreover, his Sahabah (Companions, may Allah be pleased with them) reported all the Prophet’s acts, sayings, and approvals. The religion is complete and perfect in respect of principles, explanation, and transmission. As for Dhikr, it is like any other act of ‘Ibadah (worship), bound by religious texts from the Qur’an and the Sunnah and not amenable to personal opinion. Whoever assigns certain acts of ‘Ibadah to be performed at fixed times and in a special way is required to provide the evidence. As to what is mentioned in the question, it is groundless and baseless in Shari‘ah (Islamic law). It is authentically reported that the Prophet (peace be upon him) said:
Anyone who introduces anything into this matter of ours (Islam) that is not part of it will have it rejected.
Accordingly, what is mentioned in the question is included under rejected actions.