We consider the numerical evaluation of integrals with respect to self-similar measures supported on fractal sets, with a weakly singular integrand of loga-rithmic or algebraic type. We show that, in many cases, the self-similarity of the measures, combined with the homogeneity properties of the integrand, can be exploited to express the singular integral exactly in terms of regular inte-grals, which can be readily approximated numerically using e.g. a composite barycentre rule. Our approach applies to measures supported on many well-known fractals including Cantor sets and dusts, the Sierpinski triangle, carpet and tetrahedron, the Vicsek fractal, and the Koch snowflake. We illustrate our approach via numerical examples computed using our IFSIntegrals.jl Julia code. This is joint work with Andrew Gibbs, Botond Major and Andrea Moiola.[-]