Let S be a finite subset of a compact connected Riemann surface X of genus g ≥ 2. Let M lc (n, d) denote the moduli space of pairs (E, D), where E is a holomorphic vector bundle over X and D is a logarithmic connection on E singular over S, with fixed residues in the centre of gl(n, C), where n and d are mutually corpime. Let L denote a fixed line bundle with a logarithmic connection D L singular over S. Let M ′ lc (n, d) and M lc (n, L) be the moduli spaces parametrising all pairs (E, D) such that underlying vector bundle E is stable andbe the Zariski open dense subset such that the underlying vector bundle is stable. We show that there is a natural compactification of M ′ lc (n, d) and M ′ lc (n, L) and compute their Picard groups. We also show that M ′ lc (n, L) and hence M lc (n, L) do not have any nonconstant algebraic functions but they admit non-constant holomorhic functions. We also study the Picard group and algebraic functions on the moduli space of logarithmic connections singular over S, with arbitrary residues.