Within Orthodox communities bicycle riding is discouraged on Jewish holidays for halachic reasons while Conservative communities discourage bicycling in such instances for societal reasons. (see car response)

Riding itself doesn't involve any melakha.(1) Exerting force on a pedal and through a mechanical connection moving or stopping the wheels is not a melakha of Shabbat. Halacha addresses numerous examples of similar permitted devices. Consider exerting pressure on string which, through a pulley, opens the curtains in a room. 

The five common halakhic issues raised regarding bicycle riding on Shabbat and Yom Tov follow. 

The First Issue

The primary reason bicycle riding is prohibited on Shabbat is because it enters the category of carrying (moving an item from one domain to another or from one place to another in the public domain).  Although this is permitted on Yom Tov, there is no acceptable opinion permitting it on Shabbos.  Nevertheless there are numerous communities which have eruvs which are large enough to enclose an area of a scale which would justify bicycle transportation.  The Upper West Side and Brooklyn (check with your local authority) in New York, and the Toronto community in Ontario Canada are a few examples with which I am acquainted.  By a more lenient opinion if one must reach a certain distance due to dire need it can be acceptable for a person to ride his or her bike out from private to public property and visa versa, and not walk it. Also a person may not carry anything in their pockets, etc.

As Michael and Abby Pitkowsky wrote, (M-J 26:75) R. Yosef Hayyim from Baghdad permits riding a bicycle on Shabbat for anyone when there is an eruv and when there isn't, he says when someone is needed by the public, e.g. to be a shaliah tzibbur, to hear kaddish, or to read torah, it is permitted even without an eruv (Rav Pealim, vol. 1 Orah Hayyim, no. 25).  One should look at the response there, because there are many issues which he deals with, and many conditions. 

The Second Issue:

Perhaps something on the bicycle will break and the rider will repair it. Repairing is surely prohibited. 

Rabbi Yosef Hayim (The Ben Ish Hai) in Rab Pealim I;25 (1901), referring specifically to a bicycle, states:

We should not make new gezerot based on our own opinions; it's sufficient if the people of these generations are careful with the gezerot explicitly specified by the rabbis [of the Talmud]. Therefore it is permitted [to ride the bicycle] in the Eruv on Shabbat or on Yom Tov even if only for recreation.

In the Addenda, at the beginning of Rab Pealim, also referring specifically to a bicycle, he elaborates further:

I have heard some say that we should forbid [the bicycle on Shabbat in the Eruv] because it may break and the person may come to fix it. This is also a vain argument and unworthy of being stated. First of all, it is not so vulnerable to breaking. Furthermore, we should not make decrees that weren't made by the rabbis of the Talmud, for if we do there are numerous permitted items which are vulnerable to breakage which we would have to prohibit. We find in Shulhan Aruch Chapter 339 things prohibited on Shabbat by the rabbis of the Talmud because one may come to write or fix and still we do not apply those gezerot from our own opinion to items not specifically mentioned by the rabbis. Thus, there are many cases where these gezerot that one may come to write or fix could pertain that are permitted because the rabbis didn't apply this gezera to those cases.

Hacham Obadia Yosef cites this view of Rab Pealim favorably.

If a bicycle does break while away from home on Yom Tov, it is permitted to walk it home. A flat tire or disconnected chain does not render the bicycle `mukseh' as it is fit for some use even while in its dysfunctional state, namely, sitting upon. 

Of course it is HIGHLY inadvisable to ride on Yom Tov, a bicycle whose chain comes off frequently.  A person should always check a bicycle for potential mechanical failure BEFORE the onset of Shabbos or Yom Tov.

The Third Issue:

Perhaps the rider will go beyond the tehum (2000 amot beyond the last house of the town) which is prohibited.

The same objection that we don't make our own gezerot these days applies here also. A further consideration is that the prohibition of tehumim, according to the accepted halacha, is rabbinic. The Talmudic rule that `a gezera is not made on something which itself is a gezera' applies. This latter objection is also cited favorably by Hacham Obadia Yosef.

Additionally, in a large metropolitan area such as Brooklyn, it is hardly ever the case that someone would travel beyond the tehum. 

The Fourth Issue:

Riding over an earthen surface may make a furrow in the ground that is considered in the category of digging, a prohibited melacha.

This issue is based on a gezera not to drag very heavy items over our fields for fear that we might `dig' an agricultural furrow. However, a bicycle doesn't dig a furrow; the wheels press and pack the earth rather than dig into it. Furthermore, the rider has absolutely no intention to make a furrow in the ground. Even if a bicycle necessarily makes a furrow, thus cancelling out the lenient consideration of lack of intention, the rider doesn't care about it, especially as its not a field standing for his agricultural cultivation. Where `digging' is done in such an `abnormal' manner (without shovel or spade), absent any intention for the digging and lacking any benefit from it, there surely is no prohibition. 

These concepts are developed at some length by Hacham Obadia Yosef.

For most people living in an urban environment, the chance is small (and becoming smaller each year) that you will be forced to ride off paved areas.  Most people who choose to ride a bicycle would use a bicycle path or the street.

The Fifth Issue:

Bicycle riding as a weekday activity.

Most people in our community don't ride their bicycles primarily to shop or to do other types of work [yet]. A bicycle is ridden for pleasure, recreation, to visit family and friends, etc. Indeed, for many it takes the place of the automobile and is specifically used when the latter is prohibited.

Hacham Obadia Yosef didn't think this question has much merit either.

Caf Hahayim, Rabbi Sassoon and Rabbi Obadia Yosef

The Caf Hahayim O.H. 404;8 (c. 1906), upon citing the lenient opinion of Rab Pealim in some detail, tacitly indicating it as the standard accepted halacha, states that `some' poskim were strict on this matter even though they knew that technically it was permitted. They obviously were making a gezera for their own local situation which they understood better than anyone else. Clearly the Caf Hahayim is not citing `some poskim' to indicate a universal gezera.

I asked the son of Rabbi Sassoon a"h his father's view on this matter. Following is Haki Sassoon's response:

In Letchworth we had an Eruv around our yard and we asked our teacher, Hacham Yoseph Doury a"h, a wise, learned and reliable person, if we could ride our bicycles within the Eruv on Shabbat. He told us that it is permitted providing we removed the horn and the batteries for the light before Shabbat. My father, a"h, agreed with him but was concerned with the feelings of the Ashkenazic neighbors who might not understand. So we refrained from riding on Shabbat.

Hacham Obadia Yosef, after demonstrating that none of the reasons to prohibit are adequate, states that it is proper to be strict because of the many rabbis that were strict. As he doesn't find any prohibition, and makes it clear that he decided this matter out of deference to the judgment of the majority of recent authors who published on this topic, community rabbis are not bound to modify established community custom (even if they may be inclined to do so in matters which he decides upon the actual substantive merits of the case). 

The Custom In Our Brooklyn Community

There are many areas in which it is advisable to be strict. Sometimes the rabbis will establish a takana or gezera; sometimes they will advise individuals that it is preferable to be strict privately. On the other hand, the Talmud and the great poskim throughout the centuries have pointed out that in many areas strictness may be counterproductive and have forcefully opposed strictness that is not clearly calculated to bring all-around benefit.

In our Brooklyn Syrian community the leading rabbis throughout the years have acknowledged that there is no prohibition riding a bicycle on Yom Tov. Chief Rabbi Jacob S. Kassin and others publicly quoted Rab Pealim on this many times. It is well known that a number of learned men, very particular in fulfilling mitzvoth, permitted, and when necessary, encouraged, grown members of their families to ride bicycles on Yom Tov. Never did our community rabbis declare a gezera on this matter and neither did a prohibitive minhag arise. 

Additional considerations not to prohibit what is permitted in cases such as bicycle riding on Yom Tov include the following:

* There are situations where bicycles on Yom Tov serve an important mitzvah purpose. Consider the case of an elderly couple I know, whose children, very religious, live a forty minute walk away and never used to visit on Yom Tov. When they found out a bicycle is permitted on Yom Tov they began to visit every Yom Tov.