Why did the space shuttle use bias-ply tyres instead of radials? Aircraft and automobile tyres can be divided into two main types, based on the orientation of the tyres’ body plies (the layers of tough fabric that form the main structural body of the tyre); radial tyres are built with the body plies laid perpendicular to the bead and tread (so that the body plies extend straight out radially - hence the name - from the tyre bead, straight across from one side to the other at a 90º angle beneath the tread, and straight in radially to the bead on the other side of the tyre), while bias-ply (or simply bias) tyres have the body plies laid at an oblique angle.
Radial tyres are the more common type by far, for very good reason; they are stronger than bias-ply tyres and wear much more slowly.
I was, thus, rather surprised to learn that the tyres used on the space shuttle’s main landing gear were of the older, weaker bias-ply type:

Wheels and Tires
[...]
The main landing gear tires are 44.5 by 21 inches and have 16 cord layers in a bias-ply design. They are normally inflated with nitrogen to a pressure of 370 pounds per square inch (psi). The maximum allowable load per main landing gear tire is 132,000 pounds. With a 60/40 percent tire load distribution, the maximum tire load on a strut is 220,000 pounds. The main gear tires are rated at 225 knots maximum ground speed and have a life of one landing.  [Shuttle Crew Operations Manual; my emphasis.]

The space shuttle’s main landing gear would seem to be one of the worst possible places to use bias-plies instead of radials; with each tyre carrying up to 59.9 tonnes (66 short tons) and having to cope with touchdown speeds in excess of 200 knots, it would seem imperative to use the strongest tyres available.  The magnitude of the stresses placed on these tyres was so great that, while most aircraft tyres can make it through dozens to hundreds of flights before having to be changed, the shuttle MLG tyres were rated for a grand total of one use before replacement.  Additionally, with only four MLG wheels per shuttle orbiter, even a single MLG tyre failure would significantly impair the vehicle’s braking capability, which is most emphatically not a good thing for something going as fast as a landing shuttle orbiter.  Given all this, one would think that radial tyres would be the obvious choice, hands down, for the shuttle’s main landing gear, since their greater strength and slower wear would considerably decrease the chance of a tyre failure upon landing - yet the shuttles instead used bias-ply tyres.  Why?
Simply put, for the weight bearing factor bias-ply is a lighter tire, and when building a spaceship weight is the top concern. 

The reason they were thin and single use also had to do with weight, according to NASA:


  Weight: Since weight is of extreme importance, the tires are made with
  a minimum amount of tread to conserve weight, allowing for larger
  payloads. A few pounds may not seem to make much difference, but when
  you add up all of the ways to decrease weight throughout the shuttle
  it can have a significant impact.


When it comes to weight bearing capacity: 


  Load-Carrying Capacity
  
  Advantage: Bias Ply
  
  Due to their multiple layers, bias ply tires generally have the
  advantage when it comes to load-carrying capacity. Because there are a
  uniform number of plies to support the weight of a vehicle in both the
  tread face and the sidewall, bias ply tires are generally better when
  hauling heavy loads. This is why trailer tires, tractor tires, and
  heavy equipment tires are often bias ply; they can better support
  heavy loads. Radials are available with high load ratings, but the
  weight-carrying capacity requires adding many more plies to a radial
  tire, making them very stiff.


The important line here being "adding many more plies" which will also make the tire heavier (reference above note on weight). It costs in the neighborhood of $10,000 to put 1 lb into space, so if you can save a few pounds on each tire, it does not matter that they cost \$5,660 each; you save by replacing them every time if they are indeed lighter. 


  even a single MLG tyre failure would significantly impair the vehicle’s braking capability, which is most emphatically not a good
  thing for something going as fast as a landing shuttle orbiter.


The shuttle has no issue coming to a full stop with a blowout as was the case with the shuttle Discovery in 1985 during the landing of STS-51-D.