What are the advantages of single vertical stabilizer over a twin set?
Why did engineers believe the twin fins to be superior?
There doesn't seem to be any single definitive reason for selecting the number of vertical fins on an aircraft design (except, obviously, for the twin-boom designs such as the Lockheed P-38 or Fairchild C-119).  I've seen all of the following cited by one authority or another as advantages of multiple vertical fins.  (Sorry, I didn't save or remember the sources.)

Reduced hangar overhead requirement.  This is the most common reason for including total aircraft height in a purchase specification, and it is a sine qua non requirement for carrier deployment.
Also I remember reading a magazine article that supplied this reason for the triple tail on the Lockheed Constellation.

Reduced cantilever stress at the root.  This may actually save weight over the single-fin design.  It may also help keep a tactical aircraft from disintegrating under the enormous stress of combat manoeuvres.

Reduced parasite drag.  The lower half of a vertical fin can be twice as thick as the upper half.  A twin tail design effectively omits the lower half and uses the upper half twice.

For multi-engine designs, operating the rudders in the engine slipstream affords improved rudder authority at low speed.  Pilots who taxi loaded bombers into takeoff position between other loaded bombers like lots and lots of rudder authority.
Also I notice that practically all the big flying boats of the 1940's Flying Clipper era had multiple vertical fins.  Those craft had some tricky low-speed steering requirements — they not only had to make their way through a busy harbor, but usually docked and undocked without a tugboat.  I'm only guessing at the connection though.

A single vertical fin can be masked by the fuselage wake at high angles of attack.  You can see how this could be a problem for modern fighter aircraft, which look like flying tennis courts.  Traditional bomber designs with their broad wings and heavy aerodynamic loading can also generate quite a troublesome fuselage wake.

Improved position and field of fire for a tail gunner.  I don't see it myself but I've heard the claim so I'm including it here.
I concede that the tail gunner slot in the Avro appears to benefit from the absence of bracing structure.

Reduced rudder-induced roll due to a shorter rudder offset moment.  Fast tactical military aircraft have short wings, and you don't want the rudder to extend further from the roll axis than the ailerons do.  On the other hand, the rudder doesn't need to go all the way to the top of the fin, does it.

Twin fins can be canted to reduce radar return.  They usually are, although I don't understand how the designers know where the hostile radar will be.

On the A10, the vertical fins actually mask the hot engine exhaust over a large angular area.  Thus in order to aim a ground-to-air heat-seeker up the tailpipes of an A10 you have to be directly behind it.  Which means that a few seconds ago you were directly in front of it, so good luck with that.