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We investigate the interplay between the local and asymptotic geometry of aset $A \subseteq \mathbb{R}^n$ and the geometry of model sets $\mathcal{S}\subset \mathcal{P}(\mathbb{R}^n)$, which approximate $A$ locally uniformly onsmall scales. The framework for local set approximation developed in this paperunifies and extends ideas of Jones, Mattila and Vuorinen, Reifenberg, andPreiss. We indicate several applications of this framework to variationalproblems that arise in geometric measure theory and partial differentialequations. For instance, we show that the singular part of the support of an$(n-1)$-dimensional asymptotically optimally doubling measure in $\mathbb{R}^n$($n\geq 4$) has upper Minkowski dimension at most $n-4$.

Local set approximation: Mattila-Vuorinen type sets, Reifenberg type sets, and tangent sets