How to demonstrate excellent undergraduate and graduate teaching capabilities in academic position applications?
It is a great question that I have asked myself in the last several years and I'm also looking for an informed answer. In the mean while, I have come up with an idea to demonstrate my teaching abilities to potential employers: 

After graduation, I taught several graduate courses mainly on mathematical analysis, such as measure theory, harmonic analysis, C*-algebra. Based on these experiences, I have written a lecture note on C*-algebras which is available in arXiv. I thought if a serious employer really wants to evaluate my teaching abilities, nothing speaks better than my lecture notes.