Is it St. Barts, Saint Barth, or St. Bart's?
All of those are correct...in a fashion.
The official name of the island is Saint Barthélemy. It was named by Christopher Columbus for his brother Bartolomeo (Bartholomew). The locals and the francophone world refer to it as Saint Barth or St. Barth so that's what I use.
In English, people will often change it to St. Barts and you will see that used in a lot of travel guides. Even specific online forums like SBHOnline.com and St. Barths Online use St. Barts and St. Barths respectively. However, the usage of St. Bart's, though seen often, really isn't correct. Think of the U.S. Virgin Islands, they aren't known as St. John's and St. Thomas's; St. John or St. Thomas would be correct. With that application, St. Bart's really isn't ideal. St. Barts is more correct if you are going to pop that 's' on the end.
As well, you will see St. Barths bandied about. It gets confusing, so I say act like a local and use Saint Barth. Just remember that the 'h' is essentially silent as the 'th' in English doesn't make the cut in French.