When bicycles first became popular, many poskim discussed their use on Shabbat. Almost all forbade it, for one or more of the following reasons: 1) Uvdin d’chol – Riding a bicycle is a weekday-like activity, as, among other reasons, it is a mode of transportation that takes people to many places for purposes not appropriate on Shabbat.[1] 2) Bicycles often require repairs that are forbidden to perform on Shabbat, and a rider might mistakenly perform them, forgetting that they are forbidden.[2] 3) One might possibly ride outside the techum Shabbat (boundaries of travel outside the city).[3] 4) When riding on soft ground, the bicycle wheels make grooves.[4]
Although Rav Yosef Chayim of Baghdad[5] dismissed these issues and permitted riding a bicycle on Shabbat, the consensus of both Ashkenazi[6] and Sephardi poskim,[7] and the general minhag, is to forbid it. (Some say that Rav Yosef Chayim himself later changed his mind and prohibited it.) In theory, Rav Ovadia Yosef did not consider any of the halachic issues formidable, but he agreed that it is proper to refrain from riding a bicycle on Shabbat.[8] The increasing popularity of electric bicycles likely makes riding any type of bicycle even more problematic (although one could make the opposite claim as well).[9]
Your question regarding children riding bicycles deserves special consideration on a number of grounds. First, most of the reasons to forbid bicycle-riding on Shabbat apply less to a child, especially to a young one. Such a child uses a bicycle as recreation, which is harder to classify as uvdin d’chol. In addition, a child is arguably less likely to leave the city on his bicycle or to fix it when it breaks.
Furthermore, when the accepted practice is to follow the stringent opinion regarding a particular practice, but there are strong grounds to claim that the practice is in truth permitted, there is halachic precedent for being lenient regarding children. For example, Rav Ovadia Yosef suggested that children might be allowed to wait less than six hours between meat and milk,[10] and he allowed them to eat cheese produced by a non-Jew without supervision in a case of need.[11] Similarly, some authorities permitted feeding children certain foods on Pesach that adults refrain from due only to a stringency on a question of chametz.[12] The Beit Yosef makes a statement along similar lines.[13]
We have seen that there is at least one important posek who permitted bicycle-riding on Shabbat for adults, and others have implied that although the minhag is not to ride them, this is probably only a stringency. This makes the question of bicycles for children a good candidate for leniency.
Contemporary poskim do take this approach, but only partially. Shemirat Shabbat K’Hilchata[14] writes that, except in places where the minhag is to be stringent, children may ride tricycles, but not bicycles. He cites two distinctions between the two: 1) Tricycles’ wheels do not have an inflatable tube, and one of the reasons to forbid bicycles on Shabbat is the propensity or their inflatable tubes to require fixing. 2) A tricycle is clearly a form of recreation, as opposed to being a vehicle of serious transportation.[15] It is likely that the Shemirat Shabbat K’Hilchata also factored in the fact that compared to the average bicycle rider, tricycle riders are usually much younger, which, as we noted above, makes leniency more appropriate on multiple grounds.
In summary, conventional Orthodox wisdom has determined that bicycles are forbidden on Shabbat – period. Therefore, we are not open to leniency based on age alone. However, in the separate but related case of tricycles, poskim have added up the halachic indications in a manner that permits their use.
Since a large part of the prohibition of bicycles, especially for children, is based on minhag, there is no need to oppose a minhag of leniency that may exist in certain communities (more likely among Sephardim). Furthermore, in general, even if a child is violating a clear Rabbinic prohibition, one does not have to stop him or even tell his father to do so.[16] It would be legitimate, even for a father, to allow his minor child to ride a bicycle on Shabbat if it is in consonance with the local minhag.