Good question.. I`m interested in doing a consulting internship. But more interested in Asia (Hong Kong, Japan, Singapore)

Internationals are generally accepted. Bain, BCG, MCK all take them.

As for office... it varies a little bit by company but, generally speaking the way it works is this: You apply. If they like you and decide to interview you, you have to rank your top 3 office locations. (Some have you do this when you apply, some when they decide to interview, and some before the 2nd round, it just varies ..but all of them make you do it at some point.) Hiring needs differ by office, so if you want to go work in the South Bend, Indiana office, they might have 1 position. That said, they probably have 0 applicants. If you want to work in San Fran, they might hire 5 for that office, but, you also have 50 people applying. I know what you're thinking.... and the answer is no...

Its not worth trying to game the system for two reasons: First, odds are, you'll game it wrong anyway and fail. Second, full time offers are usually given by the office at which you interned, for the office at which you interned - so if you are hoping to 'sneak in' and get an internship in Des Moines, Iowa only to then turn around and take a full time job in San Fran, dream on. So, the way to do it is to simply be upfront about your choices, list them appropriately, interview knowing that some offices may end up offering better odds than others, but you can never really know ahead of time.

In general, consulting and not, people say: "Industry, Function, Location. Pick two."
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But are there many international options, e.g. the choice to intern in Hong Kong, Singapore or Japan or are all internship opportunities in the USA only?

As far as I know you can pick any three locations including international offices. Now if you don’t speak any Italian at all and Milan is your top choice, it would put a serious question on your decision making abilities unless there is a specific reason to join the Milan office….If your top choice has no opening right away (very rare though), the companies would ask you to name another office and your second choice would automatically become the first choice….one of my peer advisors friends at Columbia just got an offer from Mck in Hong Kong which was his first choice.

Well according to some consultant companies websites. Local language ability is not a necessity in some locations, but it is advantageous. This means that if you have excellent core skills, you may still get the location of your choice

I'd watch out about the offices thing. Your school will have a recruiting office that will exert a lot of gravity. At Fuqua, the firms all come from Atlanta. Some of them are very serious about getting the recruits into their own office. Yes, they say they pass your name after R1 to whichever office you preference, but they did all that work of coming to your school and they'd like to get the goods -- not pass 'em off to another office. They also get weird about fit and culture, and it can be harder to make contacts in a distant office. This is sort of a worst-case scenario, and the firms (and probably the specific offices) will probably vary.

The ones I've talked to generally say you should intern where you want to stay. That's not to say that people never switch, but I got the impression they're not too psyched about people just wanting to be someplace cool.

In fact, some of them ask you why you are interested in certain locations -- family, office specialty, job opportunities for spouse are generally accepted, but might not even win the day.

One SY here REALLY wanted to go to TX because that's where she is from and where she wants to settle. The company insisted on sending her to a different office. She told that company to take a flying leap and went with another. (Both are top firms.)