In this article, the author localizes some classical convexity and cylindrical estimates of Huisken and Sinestrari for the mean curvature flow, allowing for initial data that isn't compact, globally 2-convex, locally noncollapsed (in the sense of Andrews) or even embedded. These extensions can be very useful because there are situations where one is naturally lead to consider the mean curvature flow of noncompact surfaces whose geometries aren't necessarily uniformly globally controlled and, for instance, lend themselves to a broad extension of the mean curvature flow with surgery of many such surfaces. The method is to localize Huisken-Stampacchia iteration using cutoff functions with the relevant geometric quantities, where controlling the extra terms which arise from the cutoff certainly takes some care.