Numerous Moment of Awesome, but the one that is a cut above the rest is the Battle of Saxe-Gotha.
Shirou beats the crap out of Guiche so thoroughly that he comes back to learn how to do that.
The entirety of Shirou vs Karin. Starting with Shirou surviving the equivalent of octagon class magic (which manages to break Rho Aias), then ending with the most badass Crane Wing Three Realm this Troper has ever seen in written form.
This troper is pretty sure it should be called pentagon class magic (since in The Familiar of Zero each class of magic is twice as strong as the previous one), but that doesn't make it any less awesome.
It's something else entirely, I believe. Each class is adding another component to the spell, and increases both power and type of effects possible. For instance, triangle wind users cannot do lightning effects at all, square ones can. Karin simply fired off two square spells, designed to be used together, within seconds of each other. So it doesn't have the increase in possible functions that pentagon magic would, but it's almost certainly more dangerous.
A more subtle one, not related to combat. When Louise finally drags a confession out of Shirou that Yes, he wants to save Tabitha, Shirou proceeds to pretty much single-handedly create a Black Ops program for Tristain, all in order to do so. The Smart Guy of the Undine Knights is thoroughly amazed by Shirou's ingenuity. This is a remarkable moment quite on par with Shirou in Heaven's Feel, when he decides he wants to protect Sakura rather than his lifelong ideal, and a crowner for the author for getting it right.
Throughout the entire story, Sheffield, a Servant bearing the Mind of God class, has been progressively built as the biggest threat to Shirou personally. Each time they clash, the battle scenes grow progressively more and more awesome until Chapter 22 wherein Sheffield realizes just how powerful Shirou is. According to the author's notes, this single scene had been planned from the very beginning of the fic, some 300,000 words back.
Shirou: All my life had been: Unlimited Blade Works.