  We define a suitably tame class of singular symplectic curves in 4-manifolds,
namely those whose singularities are modeled on complex curve singularities. We
study the corresponding symplectic isotopy problem, with a focus on rational
curves with irreducible singularities (rational cuspidal curves) in the complex
projective plane. We prove that every such curve is isotopic to a complex curve
in degrees up to 5, and for curves with one singularity whose link is a torus
knot. Classification results of symplectic isotopy classes rely on
pseudo-holomorphic curves together with a symplectic version of birational
geometry of log pairs and techniques from 4-dimensional topology.
