  We study immersed surfaces in smooth 4-manifolds via singular banded unlink
diagrams. Such a diagram consists of a singular link with bands inside a Kirby
diagram of the ambient 4-manifold, representing a level set of the surface with
respect to an associated Morse function. We show that every self-transverse
immersed surface in a smooth, orientable, closed 4-manifold can be represented
by a singular banded unlink diagram, and that such representations are uniquely
determined by the ambient isotopy or equivalence class of the surface up to a
set of singular band moves which we define explicitly. By introducing
additional finger, Whitney, and cusp diagrammatic moves, we can use these
singular band moves to describe homotopies or regular homotopies as well.
  Using these techniques, we introduce bridge trisections of immersed surfaces
in arbitrary trisected 4-manifolds and prove that such bridge trisections exist
and are unique up to simple perturbation moves. We additionally give some
examples of how singular banded unlink diagrams may be used to perform
computations or produce explicit homotopies of surfaces.
