Is there a Sahih Hadeeth relating to use of perfumed cloth of musk after menses. If there is one can we use synthetic musk because nowadays the original musk is hard to find. Will using of synthetic musk purchased from any perfume shop can cause harm to the genital part?
Sahih Muslim contains the hadith
'A'isha reported: A woman asked the Messenger of Allah (ﷺ) how to wash herself after menstruation. She mentioned that he taught her how to take bath and then told her to take a piece of cotton with musk and purify herself. She said: How should I purify myself with that? He (the Holy Prophet) said: Praise be to Allah, purify yourself with it, and covered his face, Sufyan b. 'Uyaina gave a demonstration by covering his face (as the Prophet had done). 'A'isha reported: I dragged her to my side for I had understood what the Messenger of Allah (ﷺ) intended and, therefore, said: Apply this cotton with musk to the trace of blood. Ibn 'Umar in his hadith (has mentioned the words of 'A'isha thus): Apply it to the marks of blood. (sunnah.com)
which is listed as "It is recommended for the woman who is performing ghusl following menses to apply a piece of cloth scented with musk to the site of the bleeding".
Sahih al-Bukhari contains a similar hadith:
Narrated `Aisha: A woman asked the Prophet (ﷺ) about the bath which is taken after finishing from the menses. The Prophet (ﷺ) told her what to do and said, "Purify yourself with a piece of cloth scented with musk." The woman asked, "How shall I purify myself with it" He said, "Subhan Allah! Purify yourself (with it)." I pulled her to myself and said, "Rub the place soiled with blood with it." (sunnah.com)
Fatawa indicate it's recommended, but don't say it's compulsory:
The minimum that is required for ghusl is to ensure that water reaches every part of the body, even beneath the hair. But it is better if it is done in the manner mentioned in the hadeeth from the Prophet -- Islam Q&A (and here)
Shafi'i fiqh's Reliance of the Traveller describe it as part of the after-menses ghusl as follows:
(6) If bathing after menstruation, a woman uses some musk to eliminate the afterscent of blood (O: by applying it to a piece of cotton and inserting it, after bathing, into the vagina as far as is obligatory (def: (b) below) for her to wash). (N: What is meant thereby is a substance that removes the traces of filth, by any means, and it is fine to use soap.)
(b) ... the private parts of the nonvirgin woman which are normally disclosed when she squats to relieve herself.
Where "O" indicates an excerpt from Umar Barakat's commentary, and "N" indicates commentary by Sheikh Nuh Ali Salman.
A YouTube video by Shawana A. Aziz (The Striving Muslimah) says to use musk if it's available. If it's not available, use perfume, and if it's not available use soaps, etc., or just plain water. She says "...but if perfume is available and you still don't use it, then it is makruh".
I didn't find a fatwa which addresses whether or not synthetic musk is acceptable, but given that it was regarded as non-compulsory and that soap is considered as a suitable alternative, it seems reasonable to think that synthetic musk would also be suitable.
As for whether or not synthetic musk would cause women's health problems, that's a bit beyond the scope of this site. However, it seems reasonable to believe that a lot of women are using it for this purpose, and I would expect it would be widely reported if it were problematic (outside of, say, cases of skin conditions and allergies).