Is it permissible for an Orthodox woman to do horseback riding in an ordinary saddle, or does she have to use a sidesaddle? Furthermore, would she be able to use women's trousers made for horseback riding, or does she need to dress in a skirt?
Rabbi Barry Freundel (rabbi of Kesher Israel in Washington, DC) said in a Q&A that if a woman is doing an activity more modestly done in pants than a skirt, she should wear women's pants. As examples, he mentioned bicycle riding and mountain climbing. He said there no longer exists an issue of beged ish with women wearing pants.
Rav Avigdor Nevenzahl in Hanhagos uMinhagim 16 writes there is an inyan for a woman to refrain from riding a horse because of a proof from Rashi (Pesachim daf 3a Rashi Ki Amar) by a woman its written "או על הכלי אשר היא יושבת" since it is not appropriate to mention riding or pisuk reglaim regarding a woman. For this reason it is assur for women to wear pants because of pisuk reglayim aside from other issurim.
It seems riding a horse should not be done,but pants are assur,that's how I understood it,see it inside.
וראיתי בחופה שנעשתה בהיותי שמה שהוציאו את הכלה אחרי ברכו שבע ברכות והרכיבוה על הסוס ברחוב העיר וכל הקהל עוברים לפני' ברגליהם והחתן באמצע הזקנים ולפני הכלה אשר היא לבדה על הסוס
At a wedding which took pace near my residence I witnessed the following ceremony. After the seven blessings had been repeated, the bride was placed on a horse and rode through the town. The whole community went before her on foot, the bridegroom in the midst of the elders and before the bride, who was the only one on horseback;
While not explicitly saying that horseback riding is permitted, he is describing what was evidently an established custom, and he makes no objection nor mention of anyone else objecting on the grounds that women are prohibited from riding on a horse.
(Of course, there may be other details to this incident that we are not privy too.)
Very loose pants are not allowed: First, because many of them are not tznius; and second, poskim don't allow even the ones that are tznius because it would be opening a can of worms/germs etc...
Women are not allowed to horseback ride in front of men regardless, because it's not tznius. Gemorah Pesachim 3a-b.