The Shulhan Aruch (Siman 252:2) discusses submitting items to a non-Jew for repair or cleaning over Shabbat. In general, such an arrangement is permitted if the following conditions are met:
1. The non-Jewish worker is hired as a "Kablan" (on a per-task basis).
2. The Jew did not specifically instruct the non-Jew to work on Shabbat.
3. The work is not done on the Jew’s premises
However, the Shulhan Aruch (252:3) adds that such an arrangement is only permitted with items that are not recognizable as belonging to a Jew. If the items being serviced are identifiably "Jewish," and the work is being done in a public place, there is a problem of "Marit Ayin." That is, outside observers are likely to misconstrue the arrangement as illicit; for example, they may think that the non-Jew was hired as a "Sechir Yom"-a per-hour employee.
The Poskim raise the question as to what constitutes an "identifiably Jewish" item. Does the item have to be recognized as belonging to a specific Jew, or is it even prohibited if it is an item that only a Jew would own, yet cannot be traced to an individual Jew. For example, a Talit brought to a dry cleaner is definitely owned by a Jew, yet it is not known which Jew. Another example would be a car, bearing Jewish symbols or bumper stickers, at the repair garage. It is recognizably Jewish, but not associated with any one Jew.
The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933, Siman 252:25) addresses this question and cites the Tosefet Shabbat who is lenient and permits "Jewish" items to be worked on, as long as it cannot be identified as belonging to the specific Jewish owner. Although, Rav Haim Palachi was stringent, Hacham Ovadia (Yehaveh Da’at 3:17 and Hazon Ovadia, Hilchot Shabbat p. 162) rules in accordance with these lenient opinions. He also bases himself on the fact that, nowadays, the accepted practice is to engage services, such as repairs and cleaning, on a Kabalan (per task contract) basis. Therefore, there is less likelihood of arousing suspicion.
SUMMARY: It is prohibited to have a non-Jew repair or clean over Shabbat items recognizable as belonging to a specific Jew.