Officer at of with
 

-Andersen is a personnel (or whatever department) officer with the London Post.
-Andersen is a personnel (or whatever department) officer of the London Post.
-Andersen is a personnel (or whatever department) officer at the London Post.

If they are correct, which is the most common, which is the least?

They are all correct. Here in the US using "with" is probably more common and "of" is least common. The preposition "for" can also work. UK usage may be different.

However, if the employer is a military service (such as the Royal Navy) rather than a civilian agency or private employer AND Anderson is either an enlistee or a commissioned officer rather than a civilian employee, then one correctly says "in the Royal Navy".

Thank you.