The Halacha says that, in principle, it is permissible to hire a Non-Jew for a job, such as weaving, fixing or cleaning, even though he may work on Shabbat. Since he is not being paid to specifically work on Shabbat, but on a per-job basis, any work he does on Shabbat is for himself to finish faster. This arrangement is called Kablanut (contracting).
However, it is only permitted under several conditions.
First, there is no issue of Marit Ayin, in which someone might misconstrue this as a prohibited arrangement. Therefore, it only applies to movable objects which can be worked on in private. Working fields and building houses are governed by a stricter set of rules, since everyone knows whose house is being built. While there are some leniencies in extenuating circumstances, one should be strict in this issue. Also, the work must be done on the Non-Jew’s premises- not on the Jew’s property. Additionally, it must not be obvious that the item belongs to a Jew. For example, some Poskim prohibit giving a talit to a dry cleaner, if he might work on Shabbat, since everybody knows that it’s a Jewish item.
Second, it is prohibited to instruct the Non-Jew to specifically work on Shabbat. For example, a person brought laundry to the cleaners on Friday, saying he wants to pick it up on Sunday. If there was no time for the Non-Jew to do it except on Shabbat, it's like telling him to do it on Shabbat, and would, therefore, be prohibited.
However, a normal case, where the person drops the clothes off at the cleaners on a Wednesday or a Thursday and the Non-Jew tells him that it will be ready next Tuesday, meets all the conditions and is permitted. Since the cleaner has the option of cleaning on any number of days, if he does do it on Shabbat, it was for his own benefit and consideration. Also, there is no concern here regarding Marit Ayin. The suit is a movable item, brought to the non-Jewish premises and is indistinguishable from the suits of the Non-Jews.
Similarly, in a case where one hired a tailor to make a suit, even though he might do the work on Shabbat, it is permissible. It's a movable item, in the Non-Jew’s house, and it is not known to be a Jewish item. Therefore it would be permissible. However, it would be forbidden to tell the tailor to work on Shabbat. It is also prohibited to pay him by the day. If he would work on Shabbat, the Jew would specifically be paying him to work on Shabbat. This would already not be a case of Kablanut.
SUMMARY
It is permissible to engage a Non-Jew for work on a "per-job" basis, provided that there is no concern of Marit Ayin, and the Jew does not require the Non-Jew to perform the work specifically on Shabbat.
