Summary: In this paper, we prove estimates and quantitative regularity results for the harmonic map flow. First, we consider \(H^1_{\mathrm{loc}}\)-maps \(u\) defined on a parabolic ball \(P\subset M^m\times\mathbb R\) and with target manifold \(N\), that have bounded Dirichlet-energy and Struwe-energy. We define a quantitative stratification, which groups together points in the domain into quantitative weakly singular strata \(\mathcal S^j_{\eta,r}(u)\) according to the number of approximate symmetries of \(u\) at certain scales. We prove that their tubular neighborhoods have small volume, namely \(\mathrm{Vol}\left( T_r(\mathcal S^j_{\eta ,r}(u))\right)\leq Cr^{m+2-j-\varepsilon}\), where \(C\) depends on \(\eta,\varepsilon\) and some additional parameters; for the precise statement see Theorem 1.5. In particular, this generalizes the known Hausdorff estimate \(\dim\mathcal S^j(u)\leq j\) for the weakly singular strata of suitable weak solutions of the harmonic map flow. As an application, specializing to Chen-Struwe solutions with target manifolds that do not admit certain harmonic and quasi-harmonic spheres, we obtain refined Minkowski estimates for the singular set, which generalize a result of [F. Lin and C. Wang, Commun. Anal. Geom. 7, No. 2, 397–429 (1999; Zbl 0934.58018)]. We also obtain \(L^p\)-estimates for the reciprocal of the regularity scale. Our results for harmonic map flow are analogous to results for mean curvature flow we proved in [J. Cheeger et al., Geom. Funct. Anal. 23, No. 3, 828–847 (2013; Zbl 1277.53064)].