Does the stationary object deform more than the moving one after a collision? Given two identical clay disks on an air track, one is stationary and another is moving at "high" speed.  After colliding, does the stationary disk deform more than the moving one?
If it matters, the reference frame is with the stationary disk.
Edit: Please include in your answer if it matters whether or not the speed approaches the speed of light.
 A: The reference frame doesn't matter. You cannot define a reference frame the way you did. Their deformation should be the same. The reason for this is that you can always transform to a different reference frame, i.e the centre of mass frame and physics should be the same as in any other inertial frame. 
Having established this, in the centre of mass frame everything is symmetrical so its easier to see why both objects should deform the same. Both clay disks appear to be moving with the same momentum in opposite direction, thus there is no difference between them. Since this should hold in all inertial reference frames, you can see now why none of the 2 objects is special in any sense.
Addressing the comment made above: The moving disk has more Kinetic energy in the frame that you have defined. I on the other hand can equally calculate the two body's energies in the centre of mass frame where I will see both objects moving with equal but opposite momentum and their kinetic energies will be the same. Since the laws of physics shouldn't matter on the frame of reference that we chose, then both frames are valid. Hence by transforming to the CoM frame you can verify that no one of the two bodies has more energy content than the other, i.e energy content is a frame-dependent quantity.
A: They must necessarily deform the same amount, otherwise you would violate Galilean invariance -- the idea that physics works no matter what reference frame you are in.
Suppose the stationary object $A$ deformed more. While you sit still in $A$'s reference frame, watching it deform more, have your friend run alongside the moving object $B$. In your friend's frame, $B$ is stationary, and so it should deform more than $A$. But this is a contradiction -- one cannot have the intrinsic properties of an object changing depending on who is looking at the object.
A: If the point of the air track is to eliminate friction from consideration, let's make the situation even simpler:
Imagine a completely empty universe.  Total blackness, no objects, no light, nothing.
Now add two clay lumps, and an invisible observer (you).
One clay lump appears (to you) to be motionless.  Another clay lump appears (to you) to be moving towards the first clay lump at high speeds (feel free to make this speed as close to the speed of light as you like).
Now ask yourself these three questions:


*

*If you (the observer) were to zoom over to the 2nd clay lump as it approaches and match its speed and direction so you were gliding next to it, how would the 2nd clay lump appear to be moving to you?

*How would the 1st clay lump then appear to be moving to you?

*If you were suddenly put into this situation, how would you tell if you were floating next to clay lump 1 or 2 (assuming they look identical)?
That should give you your answer.
