Ordinary generating series of multiple harmonic sums admit a full singular expansion in the basis of functions $\{(1-z)^α \log^β (1-z)\}_{α ∈ℤ, β ∈ℕ}$, near the singularity $z=1$. A constructive proof of this result is given, and, by combinatoric aspects, an explicit evaluation of Taylor coefficients of functions in some polylogarithmic algebra is obtained. In particular, the asymptotic expansion of multiple harmonic sums is easily deduced.