The state of the art in optical biosensing is focused on reaching high sensitivity at a single wavelength by using any type of optical resonance. This common strategy, however, disregards the promising possibility of simultaneous measurements of a bioanalyte's refractive index over a broadband spectral domain. Here, we address this issue by introducing the approach of in-fibre multispectral optical sensing (IMOS). The operating principle relies on detecting changes in the transmission of a hollow-core microstructured optical fibre when a bioanalyte is streamed through it via liquid cells. IMOS offers a unique opportunity to measure the refractive index at 42 wavelengths, with a sensitivity up to～3000 nm per refractive index unit (RIU) and a figure of merit reaching 99 RIU?1 in the visible and near-infra-red spectral ranges. We apply this technique to determine the concentration and refractive index dispersion for bovine serum albumin and show that the accuracy meets clinical needs.