ARCHIVED ASK THE RABBI
There are two potential issues at play in this question. The first is whether a retainer is considered a type of medicine or medical treatment prohibited under the category of refuah on Shabbat. The second is whether a retainer may be worn if there is no eruv, or if wearing it is considered carrying?
Concerning the refuah issue, the Shemirat Shabbat ke-Hilchatah says the only type of treatment that falls under this category are those treatments that can be treated with medicine. The prohibition against medicine is derived from a decree (gezeirah) that someone may grind the plants and herbs required for the medicine. In ancient times, this is how medicine was prepared. Since grinding is one of the 39 av melachot, this gezeirah was instituted. Since there is no medicine that could achieve the same results as a retainer, there is no problem of refuah.
Concerning carrying, the Shemirat Shabbat ke-Hilchatah also says there is no concern of carrying a retainer. This is because is essentially an article of clothing. It is custom designed to fit in one's mouth. It is also something that many people in our society wear. One can therefore wear a retainer even in places where there is no eruv.
Conclusion: There are no halachic problems with wearing a retainer on Shabbat.