Unexpected surfaces singular on lines in $\mathbb{P}^3$

We study linear systems of surfaces in $\mathbb{P}^3$ singular along general lines. Our purpose is to identify and classify special systems of such surfaces, i.e., those nonempty systems where the conditions imposed by the multiple lines are not independent. We prove the existence of four surfaces arising a(projective) linear systems with a single reduced member, which numerical experiments had suggested must exist. These are unexpected surfaces and we expect that our list is complete, i.e. it contains all special linear systems of affine dimension $1$, whose projectivisation has one, reduced and irreducible member. As an application we find upper bounds for Waldschmidt constants along certain sets of general lines.