IIRC, Harry Kim specifically chose Voyager, and could choose because he was near the top of his class. The episode where he slips into a parallel dimension where he was still in San Francisco and not on Voyager had, I think a friend of his didn't drop out Senior Year, so the friend was above Harry in the class rankings, and the friend chose Voyager.
Personal Opinion: Almost nobody gets a blanket choice. That is, they almost never say "We'd like to make you a Commander, which of the positions for a Commander would you like."
FAR more common is a
limited choice: "That promotion comes with a transfer, Lieutenant. The USS Hood needs a Science Officer, USS Wiffle needs a Chief Engineer, and Admiral Volstok needs a Personal Assistant. Do any of those appeal to you?"
And of course, there are those rare individuals that get a
very wide range of choices: "Jean-Luc, old friend, it's really time we got you back in command of a starship. You've seen the list of openings, is there one I could talk you into?"
"Well, I like the looks of this new Galaxy Class, and I understand one of them is to be named Enterprise."
