Casting - ClassCastException
Arrays - ArrayIndexOutOfBoundsException
Vector, ArrayList, HashMap, etc. - I rarely see exceptions when working with Java collections, but very occasionally ConcurrentModificationException
IO (File class, streams, filters, …) - FileNotFoundException
Object Serialization - ClassNotFoundException
Threads (wait(), sleep(), etc.) - In my experience, usually threading problems manifest themselves in random ways that aren't exception specific. Having to deal with InterruptedException occupies a lot of time, though I haven't seen the exception actually thrown much.
or anything else that is considered "basic Java" - by far the most common exception in my experience is NullPointerException.