The author continues his study of the gradient flow of the Möbius energy introduced by J. O’Hara [Topology 30, No. 2, 241–247 (1991; Zbl 0733.57005)], see [S. Blatt, Math. Ann. 370, No. 3–4, 993–1061 (2018; Zbl 1398.53071); Calc. Var. Partial Differ. Equ. Journal Profile 43, No. 3–4, 403–439 (2012; Zbl 1246.53086)]. The author shows a fundamental \(\epsilon\)-regularity result that allows to bound the infinity norm of all derivatives for some time if the energy is small on a certain scale. This result enables him to characterize the formation of a singularity in terms of concentrations of energy and allows the construction of a blow-up profile at a possible singularity. Finally, the author settles these problems for planar curves with the following theorem: Let \(\gamma_0\in \mathbb{R}\) be a closed smoothly embedded curve. Then the negative gradient flow of the Möbius energy exists for all times and converges to a round circle as time goes to infinity.