What countries are not apart of any continent?

Answer 1

Do a Wiki search for "List of island countries".  

Answer 2

If you go with "Oceania" rather than the old "Australia" no nation isn't on a continent. But some are transcontinental in that they sit on more than one continent; Turkey, Russia, et al. And there are some territories way out in the oceans that while technically part of continents aren't anywhere near the mainland of those continents*.

* See the most isolated island of Tristan da Cunha. Technically a British territory and far far from any land, it's technically part of Africa.  

Answer 3

New Zealand is one.   Iceland would be another .  Japan.  There are several other island countries.