Cloverfield was nominated for two 'Saturn' awards by the The Academy of Science Fiction, Fantasy and Horror Films.
Cloverfield was nominated for Best Science Fiction Film, and Lizzy Caplan was nominated for Best Supporting Actress.
It's not really clear how Cloverfield is even eligible, since it was a 2008 movie, but let's not look a gift horse in the mouth :)
* Thanks to rjlvk (?) for emailing me this!