Should any parts of Little Italy in New York City be avoided more than others?

The more south you get into Little Italy, the more sketchy it can be at night. I'm not a huge fan of the neighborhood as it comes into the border territory of Chinatown; but, the rent, naturally, decreases the further south you go. All of Little Italy is quite loud and most of the apartments are small and now expensive. I would stay North -- i.e. above Grand -- if you want to ensure safety.