Summary: Let \(P\) be a normal singularity of multiplicity \(d = 2\) or \(3\) of a complex surface \(X\). It is well-known that \(X\) is locally an irreducible finite cover \(\pi : X \to Y\) of degree \(d\) over a smooth surface \(Y\), and the singularity \((X, P)\) can be resolved by the canonical resolution \(X_k \to X_{k-1} \to \dots \to X_0 = X\), which is the pullback of the embedded resolution of the corresponding singularity \(p = \pi (P)\) of the branch locus. Let \(F\) be the maximal ideal cycle of this resolution. We will prove that \(F\) has a unique decomposition \(F = Z_1 + \dots + Z_d\) with \(Z_1 \geq Z_2 \geq \dots \geq Z_d \geq 0\), where \(Z_i\) is a fundamental cycle or zero. We show that \(w = p_a (Z_1) + \dots + p_a (Z_d)\) is an invariant of \((X, P)\) that can also be computed from the multiplicity of the branch locus at \(p\). \((X, P)\) is a rational singularity iff all of the singular points in the canonical resolution satisfies \(w \leq d - 1\). In order to get the minimal resolution from the canonical one, we need to blow down some exceptional curves, the number of blowing-downs is exactly that of fundamental cycles \(Z\) in the canonical resolution satisfying \(p_a (Z) = 0\) and \(Z^2 = -1\).