Just about ever cutter out there is unique and awesome. Now, I'm graduating in just over a year, so this has been on my mind quite a lot. I haven't seen all the different types, but I can give perception on a few.
What will make the most of any ship though, is the crew. The crew will make or break your opinion about a ship. I don't care if you have the most advance/coolest ship out there, if you have a crappy crew that you don't get along with, you will not enjoy it. Luckily, just about every crew in the CG is awesome.
Most of my experiences have either been on small boats(summer at station Annapolis on 25s and 41s) as well as one day visits (210 Mohawk, 110 Kodiak island), and even a week on a retired ship (USCGC INGHAM, 327). Out of designs, the 327 was a really cool ship, and I heard the best stories of them.... and I guess that wouldn't really count since we don't use them anymore. So, with some prejudice, I would say that the most unique ship is the Eagle.
Although, I didn't like the trip across the Atlantic on it, it is a unique ship. Once you are on Eagle, you won't forget it.
Remember though, they are designing some new ones soon.