There is no prohibition to have your kaylim work on Shabbos (as there is for your animals). However, you may not rent anything, kaylim or otherwise to a non-Jew for only the day of Shabbos since the revenue is deemed as s’char Shabbos. (ShA1 – MB1)
Even if the rental covers more than just the Shabbos day (e.g. a whole week), you may not rent out to a gentile on Friday those kaylim that are used with melacha. The reason is that if you would stipulate to him not to use it on Shabbos he wouldn’t pay you as much, so we view his melacha with the k’li on Shabbos as a sh’lichus for you. However, on any other day of the week, it doesn’t look so bad. (ShA1 – MB1,2)
Renting out tuxedos and the like, kaylim that no melacha is done with them, is allowed even on Friday. (MB2)
Lending any k’li to a gentile for Shabbos is permitted as long as it’s lent on a weekday. But don’t lend it on Shabbos or too close to Shabbos, because when people see the non-Jew carrying your possessions in the street from your house on Shabbos, they will think that you told him to carry it out – which is a melacha. (What is “too close to Shabbos”? After sunset.) (ShA2 – MB8,9)
If you and the gentile live inside a walled city where you are allowed to carry out to the street, you are then allowed to lend him even on Shabbos. However the kaylim must be of the type that no melacha is done with them. (MB9)