Examples of non-constant morphism of curves

Question

Let $f:X \to Y$ be a non-constant morphism between integral curves. Suppose further that $Y$ is non-singular. Is it possible that $X$ is singular?

Answer 1

Of course this is possible. For instance, take a projective embedding $X \subset \mathbb{P}^N$ of any singular curve. Then a general linear projection $\mathbb{P}^N \dashrightarrow \mathbb{P}^1$ gives a morphism $X \to \mathbb{P}^1$ to the smooth curve $\mathbb{P}^1$.

Answer 2

If $X$ is affine there are many non-constant regular function $f\in \mathcal O_X(X)$ and they all yield regular maps $X\to \mathbb A^1$.
If $X$ is not affine it is projective and you can use Sasha's answer.