Assalamualaikum ustaz. What is the ruling for a woman who wears perfume then went out of her house to go to the market, work and others? Hope for an explanation.
Waalaikumussalam wrt. wbt.,
Alhamdulillah, praise and gratitude to Allah SWT for His countless blessings for us all. Praise and salutations to our beloved Prophet Muhammad PBUH, his family, companions, and all those who follow his footsteps until the Final day.
According to Kamus Dewan, fragrances mean anything fragrant, smelling pleasant and delightful. 
Ibn al-Qayyim said: Perfumes that are fragrant are the food for the soul, while the soul is the catalyst of strength. And strength can be induced with the help of fragrances. It is beneficial to stimulate the brain and heart as well as other internal organs. Fragrances are able to make a heart happy, calm the soul and refresh one’s inner spirit. Fragrances are aromas that are most compatible with the soul. The relationship and bond between the soul and fragrances are very close. Fragrances are most the favoured matters of this world by the Prophet PBUH. 
He also stated: “Fragrances are something favoured by Rasullullah PBUH. Its influence is great in terms of health and prevents various forms disturbances of illnesses and factors of diseases due to its natural strength.” 
Fragrances for women
There is a hadith that can be used in the discussion of this issue. It is a hadith from Abu Mura al-Asy`ari RA, where the Prophet PBUH said:
أَيُّمَا امْرَأَةٍ اسْتَعْطَرَتْ فَمَرَّتْ عَلَى قَوْمٍ لِيَجِدُوا مِنْ رِيحِهَا فَهِيَ زَانِيَةٌ
“Any woman who puts on perfume then passes by people so that they can smell her fragrance then she is an adulteress.” 
Imam al-Munawi stated that what is meant in the above hadith is a woman who puts on perfume and its fragrant is strongly smelt then went out of her house passing through a group of ajnabi men when it is her intention to do so, then the woman is an adulteress in terms of her sins. 
Next, al-Imam al-Munawi said: “And if a woman wears perfume and passes through a ceremony (where there are men) and she induces their desires and attracts them to look at her, then every eye that looks at her has committed adultery. The woman will receive her sin for attracting others to look at her and disturb the hearts of men. Hence, it is the reason the eye adultery happens and she is included as an adulteress.” 
In another hadith from Abu Hurairah RA, where one day he met a woman who was wearing perfume and heading for the mosque. He said: “O slave woman of the Compeller, where are you headed?” She said: “To the mosque.” He said: “And have you put on perfume for that?” She said: “Yes.” He said: “I heard the Messenger of Allah (ﷺ) say:
أَيُّمَا امْرَأَةٍ تَطَيَّبَتْ، ثُمَّ خَرَجَتْ إِلَى الْمَسْجِدِ، لَمْ تُقْبَلْ لَهَا صَلَاةٌ حَتَّى تَغْتَسِلَ
‘Any woman who puts on perfume then goes out to the mosque, no prayer will be accepted from her until she takes a bath.’” 
Likewise, when considering the athar of the companions, they emphasized this issue as narrated during the time of the rule of Umar al-Khattab RA, a woman went out wearing perfume and Umar smelled her perfume, Umar hit her with his staff and said:
تَخْرُجْنَ مُتَطَيِّبَاتٍ، فَيَجِدُ الرِّجَالُ رِيحَكُنَّ، وَإِنَّمَا قُلُوبُ الرِّجَالِ عِنْدَ أُنُوفِهِمْ، اخْرُجْنَ تَفِلَاتٍ
“You women went out of your houses wearing fragrances that men can smell?! Indeed, the hearts of men are controlled by the scent they smelled. When going out, don’t wear any fragrances.” 
Is it completely impermissible for a woman to wear fragrances?
When considering the above hadiths and narrations. The prohibition of women wearing fragrances out of their houses is clear. As for the type of fragrance meant here is the one with an apparent and strong smell that attracts the gaze of ajnabi men towards her. It can induce the desires of men where their hearts are already sick and it could lead to other more harmful matters.
If this is the situation, then the act is prohibited and al-Imam Ibn Hajar al-Haithami said that it is included as a major sin even if the husband permits her to wear it. 
As for wearing a little bit of fragrance to avoid body odour such as wearing deodorant, talcum powder, soap and others, this isn’t categorized as the one mentioned in the previous hadith of the Prophet PBUH.
The reason is there is a hadith where the Prophet PBUH said:
إِنَّ خَيْرَ طِيبِ الرَّجُلِ مَا ظَهَرَ رِيحُهُ وَخَفِيَ لَوْنُهُ وَخَيْرَ طِيبِ النِّسَاءِ مَا ظَهَرَ لَوْنُهُ وَخَفِيَ رِيحُهُ
“Indeed, the best fragrance for men is what’s scent is apparent and its color is hidden, and the best fragrance for women is what’s color is visible and its scent is hidden.” 
Al-Hasan said the possibility meant with fragrances for women when they leave their houses is a fragrance with clear color but vague and unclear smell such as saffron. As for when she is with her husband (not in public) then she can whatever fragrance she favours.
Likewise, it is also stated by al-Imam Badruddin al-‘Aini: “The best fragrance for a woman is by its color and not its smell, for the color is an adornment for them while the smell if spread (apparent and strong) will lead to harm and there is concern that she’ll encounter fitnah.” 
Imam al-Nawawi said: In this hadith (the above-stated hadith), it is explained that the mustahab (sunnah) is to wear perfumes for men the same as it is mustahab for women. However, for men, it is sunnah to wear perfumes that have an apparent fragrance and vague color. Whereas for women, when she wants to go to the mosque or others, it is disfavoured for them to wear strong perfume. As for men, the encouragement to wear strong and apparent perfumes is further encouraged on Fridays and during Eid, which means when the Muslims gathered. The same for when he is attending dhikr ceremonies or classes or when he wanted to have intercourse with his wife and others. 
In our opinion, it is permissible for a woman to wear a little bit of perfume that doesn’t have a strong or apparent fragrance. This is to maintain her appearance when she is in public. It is also permissible to wear powder, deodorant, soap and others.
Avoid using strong fragrances that can attract the attention of ajnabi men that can lead to fitnah and other valleys of sin. This is as stated in the hadiths that equate such a woman to an adulteress.
 See Zad al-Ma’ad, 4/256
 See Zad al-Ma’ad, 4/308
 Narrated by Abu Daud (4173), al-Tirmizi (2786), al-Nasa’ie (5126), Ahmad (19711) dan Ibn Hibban (4424). Syeikh Syu’aib al-Arnaut stated the isnad is strong in takhrij Sahih Ibn Hibban (4424)
 See Faidh al-Qadir, 3/147
 See Faidh al-Qadir, 5/27
 Narrated by Ibn Majah (4002). Syeikh Syu’aib al-Arnaut stated its sanad is weak because there is `Asim bin Ubaidullah.
 Narrated by Abdul Razak di dalam Musannafnya (8107)
 See al-Zawajir, 2/71
 Narrated by al-Tirmizi (2788). Evaluated as sahih by al-Albani.
 See Nukhabu al-Afkar fi Tanqih Mabani al-Akhbar, 9/97
 See Syarh al-Nawawi ‘ala Sahih Muslim, 15/10