A short account on Gaussian quadrature rules for integrals with logarithmic singularity, as well as some new results for weighted Gaussian quadrature formulas with respect to generalized Gegenbauer weight $x\mapsto|x|^\gamma(1-x^2)^\alpha$, $\alpha,\gamma>-1$ on $(-1,1)$, which are appropriated for functions with and without logarithmic singularities, are considered. Methods for constructing such kind of quadrature formulas and some numerical examples are included.