Abstract: I will recall some background facts on the problem of polynomial and rational approximation of functions with singularities. Then I will report the results of my recent work with Dmitri Yafaev (University of Rennes-1). We consider functions omega on the unit circle with a finite number of logarithmic singularities. We study the approximation of omega by rational functions in the BMO norm. We find the leading term of the asymptotics of the distance in the BMO norm between omega and the set of rational functions of degree n as n goes to infinity. Our approach relies on the Adamyan-Arov-Krein theorem and on the study of the asymptotic behaviour of singular values of Hankel operators.