What is the ruling on weekly I’tikaaf, that is some people (Tableeghi) sleep in the masjid every Thursday night believing it is the Sunnah of Prophet Mohammed (salallaahu alaihi wasallam)?

The only I’tikaaf established in the life of Prophet Mohammed (ﷺ) is clear and undisputed and it is the I’tikaaf that he (ﷺ)  did in the month of Ramadan.

On one occasion he did not observe i’tikaaf during the last ten days, so he made it up in Shawwaal and observed i'tikaaf during the first ten days thereof. This was narrated by al-Bukhaari and Muslim. In the year in which he died, he observed i'tikaaf for twenty days. [1]

So, to believe in a weekly I’tikaaf to be a Sunnah is incorrect.

And Allah knows best.

Reference:

[1] Bukhaari, 2040