What's the difference between being at someone's service, of someone's service, and in someone's service?
This sentence should help clarify the differences between these:
The butler Mr. Dawkins, previously in the service of Lord Halsey as his valet, and having been of service to Lady Bucket by helping her find her lost Persian cat, said to Lord Bennett, "I am at your service, sir."
To be in (someone's) service means that you are employed by that person in some kind of service role.
To be of service (to someone) means that you have assisted or will assist them in some way. It is not necessary for you to work in service to do this. Anyone can be of service to anyone else.
To be at (someone's) service means that you are offering (or someone else has offered) your (usually temporary) help to someone, in a formal and deferential manner. Again, anyone can be at anyone else's service.
The use of the word "service" in all three relates to an older meaning of the noun, related to servant, a collective term for the various professions employed by a wealthy (or at least nominally upper middle class) family to do domestic tasks. These included such jobs as butler, valet, housekeeper, maid, chauffeur, groundskeeper, footman, cook, etc.
This was much more common a hundred years ago than it is today. Still, there are still many people who work "in service" -- but who are not called "servants", which nowadays is considered demeaning. Instead they may be collectively referred to as "the help".
If you are "of service" you have helped someone in the way a servant would have done. To be "at someone's service" is to offer your help as if you were their servant. Neither is particularly negative, however, since the acts are presumably voluntary.
Note this does not include "customer service" or "food service" or any of the many jobs that interact with the public, nor does it (usually) include people who work in hotels, spas, cruise ships, and other temporary lodgings, which are considered part of the "hospitality industry". While any of these may be "of service" or "at someone's service", I think they would not describe themselves as "in service".
At least not in the US. It might be different in other countries.
If you are in someone's service, you have some sort of duty, obligation or desire to serve that person.
The expression at someone's service is pretty much an idiomatic expression of the same thing, with perhaps a slight connotation that the service is more voluntary, as opposed to being some legal obligation. The expression at your service, in particular, is common enough that it might even sound a bit hackneyed or old-fashioned.
The expression of someone's service is not really idiomatic, but you may still find it in writing, in contexts such as "We have heard of your service in West Africa."
The Ngram is rather interesting as it shows how at your service is the most common, and was much more so a hundred years ago when it was more prominently used.