Why does a rotating tire use the static, rather than the dynamic coefficient of friction? The explanation I have heard of the difference between static and dynamic friction is that static friction is stronger because bonds form when one object is put on top of another object and these have to be overcome to get the movement started.
For a rotating tire, although the point on the ground will be stationary for an instant, it would seem that bonds wouldn't have time to form. So, why isn't the dynamic coefficient of friction used?
 A: A car drives at 20 m/s. The circumference of the wheel is 2m, so the rotation rate is 10 Hz.  A reasonable percentage of the tire is in contact with the ground - maybe around $5\%$.  That would give a contact time of $5\times 10^{-3}$ s.
This is a pretty long time in molecular terms. The distance between molecules divided by the speed of light is around $10^{-18}$ s, so that's the fastest we can imagine some sort of bonding occurring.  That 16 orders of magnitude faster than the contact time.
Real chemical reactions must be slower, but with $10^{16}$ "clicks" of time for the chemistry to sort itself out, there should be plenty of time for the tire to stick.
A: I am not sure why you are rejecting the static friction on the basis on long the parts are in contact. A "bond" is not a chemical bond that might take time, but rather an interaction between adjacent molecules, or atoms. It propagates at the speed of light, so there is plenty of time for the adjacent molecules to "bond" when sufficiently close enough.
In real life though, pairing down the tire/road contact into a friction coefficient is the wrong approach. It is a non-linear contact, where the higher the normal load the wider the contact patch is and the distribution of contact pressures changes. In addition, some parts have micro sliding as only 1 point in the contact patch is truly stationary.
There is something called the "Pacejka Magic Formula" which is a well established model of a tire contact, and there are newer ones out there which minor and major refinements to it. In the end, it depends on what you want out of it, in order for you to decide what contact/traction model to use.
[ref: Magic Formula]
A: You'd use the dynamic friction if you slam on your brakes and start skidding. Until then, your tire is rolling with the road and so the tire rubber isn't moving with respect to the road. Hence with tires rolling, you use static friction.
A: We can theorize about why static or dynamic coefficient of friction is more appropriate, and why.  However, the bottom line on why we use the static coefficient is that it has been proven, experimentally, to be the correct one.
As a thought experiment, try this: 
You have a car sitting in a flat road.  What is the coefficient of friction, static or dynamic?  The answer is "Static."  Now, roll the car very, VERY slowly (like one inch per hour).  What is the coefficient of friction?  Still static.  Now roll it just a little faster and ask the same question.  Now faster.  
How fast do you have to go before the answer is that the coefficient of friction is no longer, "static?"  
The answer to that question, in my mind, is, "fast enough for something(s) in the chassis/strut/spring/shock/axle/wheel/tire/road system to excite the rubber/concrete interface enough that radial vibration modes of the rubber in the tire were strong enough to lift the wheel off the road and keep it off." ("vibro-planing" if I may be allowed to coin a word).  The speed required to do this, on a concrete surface in good repair, is probably an order of magnitude beyond the capability of any street-legal conveyance.
