Preservation of defect sub-schemes by the action of the tangent sheaf

Let X/S be a noetherian scheme with a coherent OX-module M, and TX/S be the relative tangent sheaf acting on M. We give constructive proofs that sub-schemes Y, with defining ideal IY, of points x∈X where Ox or Mx is “bad”, are preserved by TX/S, making certain assumptions on X/S. Here bad means one of the following: Ox is not normal; Ox has high regularity defect; Ox does not satisfy Serre's condition (Rn); Ox has high complete intersection defect; Ox is not Gorenstein; Ox does not satisfy (Tn); Ox does not satisfy (Gn); Ox is not n-Gorenstein; Mx is not free; Mx has high Cohen-Macaulay defect; Mx does not satisfy Serre's condition (Sn); Mx has high type. Kodaira-Spencer kernels for syzygies are described, and we give a general form of the assertion that M is locally free in certain cases if it can be acted upon by TX/S.