  We develop a refined singularity analysis for the Ricci flow by investigating
curvature blow-up rates locally. We first introduce general definitions of Type
I and Type II singular points and show that these are indeed the only possible
types of singular points. In particular, near any singular point the Riemannian
curvature tensor has to blow up at least at a Type I rate, generalising a
result of Enders, Topping and the first author that relied on a global Type I
assumption. We also prove analogous results for the Ricci tensor, as well as a
localised version of Sesum's result, namely that the Ricci curvature must blow
up near every singular point of a Ricci flow, again at least at a Type I rate.
Finally, we show some applications of the theory to Ricci flows with bounded
scalar curvature.
