Well, I'm no english major, but me thinks that among these two, neither are gramatically write.
I guess it all depends on whether "sudden" is definate or indefinate. I'd be inclined to think it is indefinate, merely indicating "a" "sudden," since there is no need to differentiate which "sudden" is being referred to. Moreover, it would not be correct to use a definate article before introducing this particular "sudden." A similar phrase would be "In an instant."
Examine this passage: "I was engulfed by AN enormous crowd of people. Suddenly, out of THE crowd, stepped A man. THE man inexplicably informed me that you needed to quit obsessing over trivial. Furthermore, he called you AN "unholy inbred dogfaced mucklicker" and that your "dastardly dingleberry-snorting ways are despicable and distasteful."
Change any one of the those articles, and the passage is no longer coherent.
Common usage is "All of A sudden," but many people use the other phrase. I think "All of THE sudden" is more akin to "I COULD care less." Of course, outside of the States, I reckon every colloquialism may be bottomside-upards.
-Biscuits