What options are available to somebody who suddenly realizes he has something in his pocket while he walks in a public domain on Shabbat?

One option relates to the Halacha of "Makom Petur" which the Gemara establishes in Masechet Shabbat (7). The term "Makom Petur" refers to a place in a public domain that is at least three Tefahim (handbreadths) above the ground, but covers an area smaller than 4X4 Tefahim. Halacha permits one to carry from a Makom Patur to another domain, and from another domain to a Makom Patur. A Makom Patur differs in this respect from the other Halachically-defined domains (Reshut Ha’rabim, Reshut Ha’yahid, and Karmelit). When it comes to the other three domains, one may not carry from one domain to the other. Depending on the situation, this would entail either a Torah violation or a Rabbinic violation, but in any event, it is forbidden. A Makom Patur, by contrast, is treated with greater leniency, and one may carry from a Makom Patur to another domain or vice versa on Shabbat.

A modern-day example of a Makom Patur is a fire hydrant, which extends higher than three Tefahim from the ground, but is smaller than 4X4 Tefahim. Similarly, the poles often situated alongside hydrants to protect them would also have the status of Makom Patur. Therefore, one who realizes he has something in his pocket while walking in a public domain should, while walking, place the object on a fire hydrant, and in this fashion he will not have committed a Shabbat violation.

If this is not possible, then one should continue walking without stopping until he reaches a private domain. Carrying on Shabbat from a private domain to another private domain through a public domain is forbidden only by force of Rabbinic enactment, as opposed to Torah law. Therefore, if a person realizes while walking in a public domain on Shabbat that he has something in his pocket, and he does not find a Makom Petur, he can at least minimize the severity of the prohibition by continuing to walk without stopping until he reaches his home. He must ensure not to stop at all along the way, for if he does, he will then be considered as having carried from a private domain to a public domain, and if he then continues, he will have carried also from a public domain to a private domain.

Summary: If a person realizes as he walks in a public domain on Shabbat that there is something in his pocket, he should ideally put the object down on a fire hydrant as he walks. If this is not possible, he should continue walking without stopping until he reaches a private domain.