Which results in complexity theory make essential use of uniformity?
I have asked many experts essentially this question, and the answer I always get is: none.  Diagonalization proofs obviously use uniformity, and these are at the heart of the time and space hierarchy theorems, as well as Fortnow-Williams type of time-space lower bounds.  As far as I know, all the other lower bounds we know of, both for complexity class separations and for data structures, seem to be non-uniform.  It would be great to hear that I'm wrong though :).