According to the majority of poskim, riding a bicycle on Shabbat is forbidden. Therefore, "a bicycle is muktza, as ... it is a kli shem'lachto l'issur." (from that answer)
However, there is a minority opinion (that of the Ben Ish Chai) that permits riding on Shabbat within an eiruv for leisure, and even outside an eiruv for a mitzvah. Obviously, a bike is not muktzah according to this opinion. A comment on the answer linked mentions:
My wife told me that in the Syrian community in Brooklyn they ride bicycles on Shabbat, and that they base it on the Ben Ish Chai's position. – Robert S. Barnes
What is the muktzah status of a bike if there are two people spending Shabbat together, one of whom holds it is forbidden (henceforth: the machmir), and one who holds it's permissible (hencforth: the meikel)? I'll present a few different cases.
- If the bike belongs to the machmir, may the meikel ride it? Does it make a difference whether or not they were together, or planning as much, at the beginning of Shabbat? (might affect muktzah status) If it is permissible, is the machmir allowed to bring it out for the meikel to use?
- If it belongs to the meikel, is it still muktzah for the machmir? As above, does it make a difference whether or not they were (planning to be) together at the beginning of Shabbat?