We produce examples of Legendrian isotopy classes in $ST^*\mathbb{R}^n$ (the space of cooriented contact elements of $\mathbb{R}^n$, endowed with its standard contact structure) such that any element of those classes has a singular (non-immersed) wave front, that is, the projection to the base of the fibration $ST^*\mathbb{R}^n \rightarrow \mathbb{R}^n$, when restricted to such a Legendrian embedding, has local singularities. Furthermore, our examples are such that the underlying Legendrian homotopy classes contain some Legendrian embeddings whose wave fronts are immersed. These examples are motivated by a question of Arnold about the removability of singularities of wave fronts by means of Legendrian isotopy. The two key points, which are of independent interest, are as follows.
(1) A duality argument which allows the question to be translated into a problem about Legendrian submanifolds of the one-jets space of the sphere, and hence to use the technology of generating families.
(2) Several independent constructions of families of functions on compact manifolds such that the ‘critical sets’ of these families are connected. For example, we prove that given two compact, connected manifolds $M$ and $N$ of positive dimension, there exists
\[f~: N \times M \rightarrow \mathbb{R}, (x,y) \rightarrow f(x,y),\quad x\in N, \ y \in M\]
such that the equation $(\partial f/\partial y)(x,y)=0$ is regular and defines a smooth and connected submanifold of $M \times N$.
