unlocking with Three
Three's unlocking policy is more detailed than the other network operators. You may read it in full by visiting their T&Cs page. For your convenience, here's the summary:
Three state that you must not permit the handset to be unlocked via any unauthorised manner. Although they don't explain what counts as unauthorised, it is likely to mean any method not approved by them.
Three states that they may have to replace your phone as an unlocking option, and that this would be with the same handset or one of a similar specification.
You must pay an "an unlocking administration charge", currently of £15.32. They also state you must pay a "Handset Unlock Fee" if you've had it for less than 12 months (though it's not clear how much this is on top of the admin charge).
The conditions they impose are that there must be no outstanding amount owing on your account, and that you must back up any information (pictures, text messages, etc) on your phone prior to getting it unlocked as they are not responsible for any loss of data in the process.
To unlock your mobile on Three you should phone 333 from a Three mobile or 08433733333 from any other phone.
Three have a good guide to network unlock procedures by phone which may be applicable even if you have a phone locked on another operator.