Firstly: 

It is mustahabb to perfume the shroud, whether the deceased is a man or a woman. This is indicated by the saheeh Sunnah. It is proven in al-Saheehayn that the Prophet (peace and blessings of Allaah be upon him) commanded the women who were washing his daughter to put some camphor or a little camphor in the last wash. Narrated by al-Bukhaari, 1253; Muslim, 939. Camphor is a type of perfume. 

Al-Haafiz said in al-Fath: 

It is said that the reason why camphor is used, although it is a perfume, is because of the angels and others who attend, in addition to the fact that it has a drying and cooling effect; it can penetrate the body of the deceased and stiffen the body, keep vermin away, prevent wastes from being expelled, and prevent the body from decomposing rapidly. It is the most effective of perfumes in that sense. This is the reason why it is put in the last wash, because if it was put in the first wash the water would remove it. But can musk, for example, be used instead of camphor? If we think just in terms of fragrance, then yes, otherwise the answer is no. It was said that if no camphor is available, other kinds of perfume may be used, even if it has only one of these qualities. End quote. 

Al-Nawawi said in Sharh Muslim: 

This indicates that it is mustahabb to use a little camphor in the final wash. There is consensus among us on this point, and this is the view of Maalik, Ahmad and a number of scholars, because of this hadeeth, and because it perfumes the deceased and makes his body stiff and cool, and prevents the body decomposing rapidly. End quote. 

It was narrated that Jaabir said: The Prophet (peace and blessings of Allaah be upon him) said: “When you perfume the deceased with incense (bakhoor) then perfume him three times.” Narrated by Imam Ahmad, 14131. al-Nawawi said in al-Majmoo’ (5/155): Its isnaad is saheeh. It was also classed as saheeh by al-Albaani in Saheeh al-Jaami’, 278. 

The word deceased here refers to both male and female. 

What is meant is perfuming the shroud with incense. Al-Bayhaqi mentioned in his Sunan (3/568) that this hadeeth was also narrated with the words, “Perfume the shroud of the deceased with incense three times.” 

See Badaa’i’ al-Sanaa’i’, 1/307. 

It was narrated that Asma’ bint Abi Bakr (may Allaah be pleased with her) said to her family: “Perfume my garments with incense when I die, then apply hanoot to me.” Narrated by Maalik in al-Muwatta’, 528; and by al-Bayhaqi in al-Sunan al-Kubra, 3/568. 

It says in al-Muntaqa: Hanoot is that which is put on the body and shroud of the deceased of perfume, musk, amber, camphor and anything else that is used for its scent rather than its colour, because the purpose here is the scent rather than beautification with the colour. End quote. 

This ruling – that it is mustahabb to perfume the deceased – does not include one who is in ihraam for Hajj or ‘Umrah, because the Prophet (peace and blessings of Allaah be upon him) said concerning a man in ihraam who died at ‘Arafah: “And do not put any perfume on him.” Narrated by al-Bukhaari, 1851; Muslim, 1206. According to another report: “Do not bring any perfume near him.” 

Al-Nawawi said: It is mustahabb to perfume the shroud with incense except in the case of a man or woman who is in ihraam. 

Al-Majmoo’, 5/156 

Secondly: 

With regard to the manner in which the deceased is to be perfumed, the perfume is to be placed on the parts of the body on which one prostrates, because of their noble status, and on the places where dirt gathers such as the backs of the knees. If the whole body is perfumed, there is nothing wrong with that. 

Al-Bayhaqi narrated (3/568) that Ibn Mas’ood (may Allaah be pleased with him) said: Camphor is to be put on the parts of the body on which one prostrates, which are: the forehead, the nose, the hands, the knees and the feet, because he used to prostrate on these parts so they are deserving of greater honour.  

See Sharh Fath al-Qadeer, 2/110 

Ibn Qudaamah said in al-Mughni (3/388): The hanoot (perfume which is used for the deceased) should be placed on the joints such as the backs of the knees and beneath the armpits, because these are places where dirt gathers. And it should be put on the parts on which he used to prostrate because they are nobler. And if the entire body is perfumed there is nothing wrong with that. End quote. 

Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) was asked: Is there any report that the entire body of the deceased may be perfumed? 

He replied: Yes, that was narrated from some of the Sahaabah (may Allaah be pleased with them). 

Majmoo’ Fataawa Ibn ‘Uthaymeen, 17/97 

Thirdly: 

If a woman who was in mourning following the death of her husband dies, should she be perfumed? 

Al-Nawawi said in al-Majmoo’ (5/164-165): 

The correct view is that it is not haraam to put perfume on her, because perfume is forbidden to her during her ‘iddah so that no one will want to marry her, but this consideration no longer applies after she died. End quote.