Even when there is an eruv permitting carrying on Shabbos, people do not ride bicycles to shul.
I know that doing this is not permitted, but why not?
The reason why we don’t ride bicycles on shabbos, even when there is an eruv is because a bicycle is used to travel long distances, and we may accidentally ride it outside of the eruv. Other reasons given include the fear that it might break, i.e. it might get a flat tire, the chain may fall off, and we will come to fix it. A tricycle though that has solid wheels and isn’t meant to ride long distances doesn’t have these issues, therefore it may be used inside an eruv.
Menucha Nechona 38-3, Tzitz Eliezer 7-30(1), Ketzos Hashulchan 110 Badei Hashulchan 16, Shmiras Shabbos Khilchoso 16-18, Tiltilei Shabbos pg. 24 ftnt 21 in the name of R;’ M. Feinstein zt”l, Mbais Halevi Tishrei 5755 pg 53, Nachlas Yisroel pg 280,