I have seen different prepositions after "Good luck". Example:
Good luck on/with/for your new job
Could you explain the possible differences of usage or meaning? Thank you.
The most common preposition paired with "good luck" is "to." The meaning is clear enough, but here are a few examples:
"Good luck to people selling rubbish plates, but for us it's nothing but bad news."
"It's just the way depression works. Meds will fix it. Good luck to you."
Next most common, according to some quick on Brigham Young University's English corpora (i.e., massive repositories of English texts) is "with." This matches my own sense of how the word is used. The preposition is used mainly with a specific thing, such as a job or plan. And of course, the sardonic "good luck with that" is popular. Some examples:
"Thank you so much for calling and good luck with making it all work."
"Good luck with, well, whichever career you decide you to follow."
"On" is very similar to "with" in meaning. I personally don't see a difference, even after looking up many examples of both words' use. One commenter suggests that "on" is used when only the result matters, whereas "with" is used when the process is more important. That doesn't seem to be true; maybe others would like to weigh in?
One clear case is when "good luck" is being combined with an existing idiom using "on." For instance, "Governor Huntsman, thanks so much for joining us. Good luck on the campaign trail." Otherwise, I'd say you can generally use with, as it is more common, and use "on" only when necessary.
Finally, "for" resembles "to" except that it's used much less frequently ("to" is about 8 times more popular) and expresses less of a wish and more of a statement of fact.
Narrowly addressing the original question, you can say "Good luck on your new job" or "Good luck with your new job" but you cannot say "*Good luck for your new job." As shown above, "for" can only be used when the expression is not a wish.
I claim the distinction between "on" and "with" is that "on" wishes luck over a process while "with" wishes it on an outcome. Consider these cases:
"Good luck on that paper." I'm wishing him success writing the paper. "Good luck with that paper." I'm hoping his teacher takes pity on him.
BUT, if he's en route to class, paper in hand, then they mean exactly the same thing because almost nothing is left of the process.
I can say "Good luck with your boss" to a person who's having trouble at work, but I cannot say "Good luck on your boss" because the boss isn't a process. I could say "Good luck on getting that raise" though.
I would claim, then, that the reason you can't say "Good luck with the campaign trail" because the trail cannot be an outcome.
I think it is a pragmatic side-effect that using "with" when I could have used "on" implies that I think the result is out of the control of the listener. Tone of voice can strengthen this effect, as in "Good luck with that."
"Good luck to" is directed at a person or thing.
"Good luck with [something or someone]" is wishing someone 'good luck' with a particular 'project'. It is sometimes used sarcastically to suggest that an event has little chance of occurring - "I want to be President of the USA." "Yeah, good luck with that."
I would not generally use, "Good luck on ...", unless it were a date or occasion. "Good luck on your birthday." "Good luck on Friday."
"Correct" usage with regards to prepositions is tricky. Depending on where you live in the country, it's equally correct to say, "Next in line" or "Next on line."
In the UK, in is sometimes used as a preposition when wishing luck: "good luck in your exam tomorrow". And to disagree with Greg Hullender's answer above, in some instances for can be used too: "good luck for tomorrow".
I would have added this as a comment but can't.
'Good luck on' is wrong in UK English.