By Günter Neessen
Does it depend on the state?
In literature and higher register, “nein" is the word of choice. Regarding everyday speech, I can't answer for whole Germany, but where I come from, the west of the northern lowlands, “nee” is much more often used than “nein". Common people in their own peer group would only use “nee".
The reason is of course that “nee" is the Low German/Low Saxon word for “nein"/”no", and while Low Saxon tends to decline in favour of Standard German, the people integrate traits of it into something one might call “Northern Standard German”.
Edit: Regarding the reactions, it seems that “nee" is widespread within the German realm, not only in the north, but even at least in Central Germany.