We provide an orthogonal basis of polynomials for the local Dirichlet space $\mathcal D_\zeta$. These polynomials have numerous interesting features and a very unique algebraic pattern. We obtain the recurrence relation, the generating function, a simple formula for their norm, and explicit formulae for the distance and the orthogonal projection onto the subspace of polynomials of degree at most $n$. The latter implies a new polynomial approximation scheme in local Dirichlet spaces. Orthogonal polynomials in a harmonically weighted Dirichlet space, created by a finitely supported singular measure, are also studied.