If you value your sanity, do not go to Leicester Square or any club on the Haymarket. Packed full of tourists and people who know nothing about clubbing. I've probably had two good nights there in the umpteen times I've visited, and that's probably because it was a private party.
Soho is good (Leicester Square is categorically outside Soho) and bad. Punk is a great little place sat between Oxford Street and Soho Square. Moonlighting is bad.
Fabric, Matter and Ministry are generally visited for the music, that's to say you will get tired and it will be unbearably loud. Don't go unless you want to get unbearably drunk or dance the night away without stopping at all. I love Fabric but I can say a night there is very tiring indeed. Heaven under Charing Cross Road is very good though it's mostly gay nights there only.
Camden is becoming more and more mainstream by the minute, regrettably. Wouldn't recommend going on a weekend but the monthly Buttoned Down Disco at Koko is amazing, and the night before I moved to France I went to an 80s night at Underworld! Proud's also okay on a weeknight.
I haven't actually ever been out in Shoreditch. I want to.