We study the effect of regular and singular domain perturbations on layer potential operators for the Laplace equation. First, we consider layer potentials supported on a diffeomorphic image ϕ(∂Ω) of a reference set ∂Ω and we present some real analyticity results for the dependence upon the map ϕ. Then we introduce a perforated domain Ω(ε) with a small hole of size ε and we compute power series expansions that describe the layer potentials on ∂Ω(ε) when the parameter ε approximates the degenerate value ε = 0.