The aim of this talk is to give several results concerning the ''size'' (from different points of view) of sets defined using ultradifferentiable classes.
More precisely, it is known that the set of nowhere analytic functions is ''large'' in the space of smooth functions; we will first show how to extend these results to nowhere Gevrey functions. Then, we will study functions which are nowhere quasianalytic or nowhere in a given ultradifferentiable class. Finally, we will present some recent results (obtained with G. Schindl) concerning the size of the image of non-surjective Borel mappings.