actually, both are NOT correct. Both are USED but that doesn;t make them correct! Only one is correct, in italian, and since this is an italian dish, it's fettuccine, with the final E.
the i, in general, is the masculine plural ending. the e is usually the feminine plural ending. And fettuccine is the plural of the feminine noun fettuccina, which is the diminutive of fettuccia, meaning ribbon.
Now when words cross borders and are used in other languages, the spelling rules often go to the winds. But unlike english, which can hardly be said to have spelling rules, Italian spelling has rules with no exceptions, except occasionally for imported words. To the extent that if you want to spell a name out loud in italian, you just pronounce it carefully. So, fettuccine is pronounced with an "eh" sound, (like the sound in "bread" - exactly like the first e in "fettuccine") at the end, not with an "ee" sound (like in "teeth")
But then again, everyone calls it fettuccini in english, and eventually it might become an english word, not an italian word. ("Zucchini" is an example of this. "Zucchina" would be the singular in italian, and the plural is "zucchine", not "zucchini" (which would be the plural if the singular were zucchino, which it isn't) But by now it's an english word (zucchini) and i've even heard it pluralized to zucchinis!
So, all said, i think if you want it to be classy, you'd want to use the true italian spelling and call them fettuccine. And if you're using "zucchini" in a dish with an english name "fried zucchini" that' s fine, but if you give the dish an italian name, like "zucchine gratinate" you should use the italian spelling.
"Siduri said, 'Gilgamesh, where are you roaming? You will never find the eternal life that you seek...Savour your food, make each of your days a delight, ... let music and dancing fill your house, love the child who holds you by the hand and give your wife pleasure in your embrace.'"