In a nutshell, Girls’ bicycling is permissible as long as a woman is modestly dressed and is sticking to the teachings of Islam in general, there is nothing wrong with her playing, exercising, speaking to men, or doing anything else in other social activities.
Answering the question, Dr. Salah Al-Sawy, secretary-general of the Assembly of Muslim Jurists in America (AMJA), stated,
There is nothing wrong with a girl’s bicycling for exercise, so long as she remains committed to her hijab and modesty. In addition, she should be distant from the eyes of any man who is not a mahram (husband or close male relative forbidden in marriage).
This fatwa is based on the general jurisprudential principle stating that permissibility is the origin ruling for things, except for those matters prohibited by Shari`ah. As far as I know, there is no evidence that prohibits this act.