In the thesis [Pe1] it was introduced, studied and applied a general theory of Weierstrass loci for vector bundles on a smooth curve. The results, proofs, background, examples and motivations of this thesis are contained in [Pe2]. We believe that this theory, at least in characteristic 0, is the ‘right’ one. The aim of this paper is to introduce and study an extension of [Pe1] to the case of higher dimensional varieties. At least two possible theories seem to be useful and natural; see the discussion just after Remark 1.1 and Section 4. We strongly prefer the ‘symmetric’ one (see Definition 1.5). In the first section we introduce the general theory and give the main general results. In the second section we study in details the case of $P^2$ for three reasons: it is nice; it shows how to use the general theory and what could be expected in more general situations and (last but not least) to convince the reader that it is technically easier and often more interesting to work in the ‘symmetric’ set up. Then in the third section we apply the method of Section 2 to a much more general situation (essentially, any variety $X$ as base of the vector bundle). In the fourth section we give the set up and start the analysis of specific examples of what happens near a specific point $P$ of the base variety $X$ (even when $X$ is singular at $P$). Here, except at the first step we are able to work only with the ‘symmetric’ definition.