I spent 3 weeks at Rood and Riddle and 3 weeks at Hagyards this spring/fall. There were a TON of international students at both places the whole time I was there. They call the extern house at Rood and Riddle "The United Nations" for a reason! 😉
You will see a ton of surgery at both places, especially if you visit in the spring. I think a few days, I saw 10-20 surgeries in a day at Rood and Riddle. I didn't actually spend any time in surgery at Hagyards, but at Rood and Riddle, it's a good mix of orthopedic/arthroscopy, soft tissue (lots of tie backs/tie forwards and cervical laceration repairs), repro emergencies, and colics.
The only other REALLY busy private practice that I'm familiar with is Mid-Atlantic in New Jersey. From what I understand from classmates who have visited, you see a lot of surgeries! I've visited a ton of less-busy private practices as well -- but if you're coming all the way over here, you may as well get your money's worth! 🙂
New Bolton is busy (especially compared to the other veterinary schools), but if you really want to see a large VOLUME of cases, Kentucky in the spring is your best bet. On the other hand, if you want more of a learning/patient-contact environment, you'd be better off going with a veterinary school.
Most of the international students seemed to group their visit -- spend 2 weeks at R&R, 2 weeks at Hagyards, 2 weeks at Woodford Equine, etc.
Private practice externships in the US tend to be on a first-come, first-serve basis. I was able to visit every practice I wanted to by planning ahead. There seems to be a kind of funny pattern to it though -- either you need to plan a year in advance, or a month in advance. Lots of people seem to cancel a week or two before they're supposed to show up somewhere! And if you're willing to pay for/find your own housing for your visit, most places are more than happy to let you visit.
I'm not sure how competitive it is to get an externship at New Bolton, honestly. We don't seem to have a ton of externs, but we definitely have them -- a lot of visiting students seem to do the NICU rotation in the spring. I think it depends on how many Penn students are enrolled in each rotation, since we get first priority. If there are still spots open in a rotation, I would imagine that they would gladly take an extern.
NCSU's equine orthopedics service is also pretty busy -- and I had no problem scheduling externships there...so that might be another option.