Praise be to Allah.
The woman is required to cover all of her body in front of non-mahram men with clothes that are loose and will not show the size of her limbs and are not see-through.
The basic principle is that it is permissible for a woman to ride animals, just as women used to ride camels and so on at the time of the Prophet (blessings and peace of Allah be upon him). al-Bukhaari (5365) and Muslim (2527) narrated from Abu Hurayrah (may Allah be pleased with him) that the Messenger of Allah (blessings and peace of Allah be upon him) said: “The best women who ride camels are the women of Quraysh.”
Some fuqaha’ forbade women to ride horses except in cases of necessity such as jihad or Hajj, and they gave as the reason for that the fact that it is an imitation of men, and because of the reports that the women who do that are cursed, but that is a hadeeth that is not saheeh.
It says in al-Durr al-Mukhtaar: A Muslim woman should not ride an animal because of the hadeeth. This applies if it is for leisure, but if it is for a need such as a campaign or Hajj or some religious or worldly interest concerning which she has no other option, then there is nothing wrong with it.
Ibn ‘Aabideen said in his Haashiyah (6/423): The phrase “because of the hadeeth” refers to a hadeeth in al-Dhakheerah which says “May Allah curse the women on saddles.” But al-Madani narrated from Abu’l-Tayyib that there is no basis for it, i.e., there is no source for the hadeeth in this wording but the meaning is proven. In al-Bukhaari and elsewhere it says: The Messenger of Allah (blessings and peace of Allah be upon him) cursed men who imitate women and women who imitate men. In al-Tabaraani it says: A woman passed by the Messenger of Allah (blessings and peace of Allah be upon him) carrying a bow and he said: May Allah curse women who imitate men and men who imitate women.
The words “but if it is for a need such as a campaign” and so on mean: it is stipulated that she should be covered and she should be with her husband or a mahram. The words “for some religious interest” refer to something like travelling in order to uphold ties of kinship. End quote.
If a woman rides a bicycle in front of men there is the possibility that she may become uncovered and part of her body may appear or the shape of her body may appear when she moves and air flows around her. For this reason it is not permissible for her to do that except in the case of necessity or urgent need, on condition that she wear clothes that cover her, with pants and socks underneath.
We asked Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) a previous question, which was as follows: Is it permissible for a Muslim woman living in a kaafir country to ride a bicycle or motorbike wearing complete hijab?
He replied: I do not think this is allowed, because she may be caught up with pursued. End quote.
Dr. Ahmad al-Hajji al-Kurdi [an expert in al-Mawsoo‘ah al-Fiqhiyyah and a member of the Ifta Committee in Kuwait] was asked:
What is the ruling on women riding bicycles in European countries in order to get to school or work or to the supermarket?
There is no reason why a woman should not ride a motorbike or a bicycle if she is where men cannot see her, so long as she adheres to complete shar‘i hijab that covers her body and she is careful to avoid showing any part of her ‘awrah when getting on and off.
But with regard to her riding it in a place where men can see her, I do not think that this is permissible except in the case of urgent need, because usually when a woman rides it, part of what she has covered appears or the clothes become tight and show the shape of her body. And because she may fall from it and uncover what she has covered, or there are other reservations.
Moreover, among the conditions of a woman’s hijab in front of non-mahram men is that it should cover all of the ‘awrah; it should be thick and not see-through;, it should be loose and should not show the shape; and it should be of a dull colour that does not attract attention; it should not have any adornment or pattern; and it should not be intended as an adornment, rather it should be worn in order to cover.
The best that I have seen in that regard is what is called the jilbaab, but I do not say it is obligatory for anyone.
End quote from his website.
And Allah knows best.