In this and this papers Thurston proves that every distribution is homotopic to an integrable one (in the first one for codimension greater than one and in the other for codimension one).

Recently, I've came up with a nice paper by Burago and Ivanov which with some other hypothesis manage to start with a distribution and construct a foliation tangent to arbitrarily small cone field around it.

I am far from foliations, but I was wondering if there are examples of distributions which are not integrable and can not be perturbed in the $C^0$ topology in order to become integrable (Edit: That is, a distribution is $\epsilon$ close to other if it is contained pointwise in a cone of angle $\epsilon$ of the original one.)

References appreciated!