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22
Answer: We have, in the past, discussed bicycles (forbidden) and tricycles (permitted), and the sources on the two can help us analyze the less common unicycle, which we have not found discussed by the poskim. We will refer to riding indoors or within an eiruv. Otherwise there are serious carrying issues (see Living the Halachic Process VI, C-12, regarding the similar but not identical case of a rickshaw).

Bemare Habazak - Rabbis Questions (591)
Rabbi Daniel Mann
545 - Switching the Location of a Mezuza
546 - Unicycles on Shabbat
547 - Whose Pronunciation Should the Oleh Use?
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Shemirat Shabbat K’hilchata (ibid.) says that children can be allowed to ride tricycles – but not bicycles. He cites two distinctions between the two (see ftnt. 53). 1) Tricycle wheels do not have an inflatable tube. 2) A tricycle is clearly a form of recreation, as opposed to serious transportation. It is also likely that he factored in that the average tricycle rider is a child.
How should we view a unicycle? A classic unicycle shares features with a bicycle, including an inflatable tire, so reason #2 to forbid bicycles applies. However, when it is used as a hobby or for non-professional exhibition, elements of uvdin d’chol and going out of techum Shabbat would not apply. (We are not referring to use for the uncommon sport of unicycles on mountain trails, where #4 could apply.) Also, unicycles did not exist when the original bicycle minhag began, and they are not used interchangeably with a bicycle. Therefore, one could argue against extending the bicycle minhag/ruling to unicycles. In Bemareh Habazak (IX:8), albeit under circumstances that include significant need, we entertained the possibility of distinguishing even between clearly different models of bicycles, based on the different likelihood of problems in one versus the other.
On the other hand, given similarities in name and design and given that some of the explanations of the prohibition on bicycles do apply to it, it is likely that poskim would not allow it, especially since the need for it on Shabbat is rarely significant. If a child under bar mitzva wanted to use it, that would be significantly more lenient because of his lower level of obligation in mitzvot, which encourages leniency (see this column, Vayeira 5777).
My basic research indicates that "unicycles" are nowadays also used for transportation, which can make the issues for bicycles of uvdin d’chol and techum Shabbat applicable. On the other hand, that is apparently mainly with electric unicycles (which are anyway forbidden because of the electric element) that are used for transportation. It is doubtful, though, that we must be more machmir due to the existence of electric unicycles, especially since their design is totally different.
In the final analysis, we do not recommend allowing unicycle riding on Shabbat, but for someone (especially a child) who uses it only for private recreation, leniency is conceivable.

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