The official phrase approved by dictionaries is all of a sudden. The phrase dates back to Shakespeare in The Taming of the Shrew, though oddly the variant all of the sudden was in print six years before Shakespeare.
The word sudden itself means at once or without warning, and all of a sudden is a long way to say suddenly, they are interchangeable. Grammatically speaking there is little difference between using the article a or the. However, dictionaries side with Shakespeare.
Another archaic version of the idiom is on a sudden.
Side note: Sudden also has the nominal derivative of suddenness.