Sard's lemma is an example - the set of regular values is non-empty since it has positive measure.

The following example also answers your question: Recently I proved the following lemma.

Let $f\colon M\to N$ be a smooth map, $M$ is a non-empty paracompact manifold. Let $k$ be a maximal rank of a differential $df(x)$ over $x\in M$. Then there exists a point $y$ in $f(M)$ such that the rank of the differential $df$ is maximal for all points of $f^{-1}(y)$.