The $\varepsilon$-regularity phenomenon shows up in several different contexts. I try to describe it focussing on the harmonic map situation, but I really would like to understand the situation in general. The following is the Schoen-Uhlenbeck $\varepsilon$-regularity lemma, extracted from [Tobias H. Colding, William P. Minicozzi II, An excursion into geometric analysis][1].

Let $N$ be a Riemannian manifold and $B_{r}$ be the ball of radius $r$ centred at the origin in $\mathbf{R}^k$. Then there exists $\varepsilon(k,N)$ such that if $u:B_{r}\in\mathbf{R}^k\rightarrow N$ is an energy minimizing map and $$\frac{\int_{B_{r}}|\nabla u|^2}{r^{k-2}}<\varepsilon,$$ then $u$ is smooth in a neighborhood of $0$ and $$|\nabla u|^2(0)\leq \frac{C}{r}.$$

Thus if a (conformally invariant) rescaling of the energy that $u$ minimizes is small (I suppose $u$ should be in a suitable Sobolev space), then $u$ is automatically smooth in some smaller ball. This rescaling is monotonically increasing thanks to a monotonicity lemma. I am not sure how to interpret the bound on the derivative at zero, though. The $\varepsilon$-regularity lemma quickly implies that the singular set $S$ of $u$ has $(k-2)$-dimensional Hausdorff measure zero.

My questions are:

1. What are the basic ingredients (I suppose I am talking about the properties of the energy functional here) that guarantee that such a lemma holds?
2. What is the meaning of the supremum of the set of all $\varepsilon$ such that the energy bound holds, and how can it be computed?
3. Is there a simple intuitive picture that I am missing that explains the situation?