"I don't have a wife", "I have no wife" and "I haven't a wife" are all correct and mean the same thing, but there are subtle differences in usage. "I don't have a wife" sounds the most neutral. "I haven't a wife" means the same thing, but it sounds (and is) very old fashioned - we usually only use "have" as a modal verb when it's part of a perfect tense in Modern English. "I have no wife" is more emphatic. Whereas "I don't have a wife" could simply mean I'm not married, I might say "I have no wife" if I'm very angry at my wife and am implying that I may be considering a divorce. Similarly "I have no son" (or Twyin's "You are no son of mine") is something you might hear a movie character say if they have a son who they have disowned. It doesn't have to be that extreme. "I had no trouble..." is basically the same as "I didn't have trouble...", except the former emphasises that I had no trouble AT ALL. Remember, this only applies when "have" isn't a modal verb. If it is, you have to use "have not". If it isn't, "have not" wouldn't be wrong, but it would be awkward.