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We investigate inverse thickness $1/\Delta$ and the integral Menger curvatureenergies $\mathcal{U}_{p}^{\alpha}$, $\mathcal{I}_{p}^{\alpha}$ and$\mathcal{M}_{p}^{\alpha}$, to find that finite $1/\Delta$ or$\mathcal{U}_{p}^{\alpha}$ implies the existence of an approximate$\alpha$-tangent at all points of the set, when $p\geq \alpha$ and that finite$\mathcal{I}_{p}^{\alpha}$ or $\mathcal{M}_{p}^{\alpha}$ implies the existenceof a weak approximate $\alpha$-tangent at every point of the set for $p\geq2\alpha$ or $p\geq 3\alpha$, respectively, if some additional densityproperties hold. This includes the scale invariant case $p=2$ for$\mathcal{I}_{p}^{1}$ and $p=3$ for $\mathcal{M}_{p}^{1}$, for which, to thebest of our knowledge, no regularity properties are established up to now.Furthermore we prove that for $\alpha=1$ these exponents are sharp, i.e., thatif $p$ lies below the threshold value of scale innvariance, then there exists aset containing points with no (weak) approximate 1-tangent, but such that thecorresponding energy is still finite. For $\mathcal{I}_{p}^{1}$ and$\mathcal{M}_{p}^{1}$ we give an example of a set which possesses a point thathas no approximate 1-tangent, but finite energy for all $p\in (0,\infty)$ andthus show that the existence of weak approximate 1-tangents is the most we canexpect, in other words our results are also optimal in this respect.

Tangency properties of sets with finite geometric curvature energies