Scalar and vector bosons in the background of one-dimensional nonminimal vector linear plus inversely linear potentials are explored in a unified way in the context of the Duffin-Kemmer-Petiau theory. The problem is mapped into a Sturm-Liouville problem with an effective singular oscillator. With boundary conditions emerging from the problem, exact bound-state solutions in the spin-0 sector are found in a closed form, and it is shown that the spectrum exhibits degeneracy. It is shown that, depending on the potential parameters, there may or may not exist bound-state solutions in the spin-1 sector. © 2012 IOP Publishing Ltd.