In many places, cycling is a very common and finance friendly mode of transport. Is it permissible for women to ride bicycles if they wear lose clothes that don’t define their bodies?
If the woman is wearing full Hijab (lose clothes that don't define her body) cycling will be permissible for her, although it is better for her to avoid it as it is mentioned in many books the it is Makrouh (disliked). Makrouh is after all permissible.
Wassalam.