A novel luminescent Cu(II)–organic framework, {[Cu(asba)(3bpmo)_0.5(H₂O)]·H₂O}_n (1), was obtained through the assembly between copper(II) and 2-amino-5-sulfobenzoic acid (H₂asba) in the presence of an auxiliary flexible ligand, N,N′-bis(3-pyridylmethyl)oxalamide (3bpmo). 1 is a 2D (6, 3) topological network, which further extends into a 3D supramolecular network via hydrogen and chalcogen bonds. An indirect bandgap exists in 1. 1 shows high framework stability in common solvents and reversible dehydration–rehydration behaviour. Remarkably, 1 is an especially rare example that exhibits novel solvent-dependent luminescence sensing properties. In DMF, 1 shows selective and sensitive fluorescence “turn-off” sensing of Fe³⁺ over a number of coexistent metal cations. 1 could also be used as a potential selective and sensitive probe for HCO₃⁻ among a variety of interfering anions. Furthermore, 1 shows discriminative sensing of HCO₃⁻ over CO₃²⁻. In water, 1 shows a quite different fluorescence sensing behaviour from that in DMF. 1 cannot sense Fe³⁺ and HCO₃⁻ in aqueous medium. It is fascinating that 1 shows selective and efficient fluorescence “turn-on” sensing of Al³⁺ accompanied by a bathochromic shift of the emission peak in water. The limit of detection (LOD) for Al³⁺ reaches the 0.80 μM (0.02 ppm) level.