In this paper, we for the first time propose a novel partitioning-based constellation design approach for discrete Fourier transform-spread-orthogonal frequency division multiplexing modulation with dual-mode index modulation (DFT-S-OFDM-DM) in visible light communication (VLC) systems. Specifically, two partitioning-based constellation designs, i.e., block-based constellation partitioning and interleaving-based constellation partitioning, are proposed to generate two distinguishable constellation sets for DFT-S-OFDM-DM in VLC, by considering four 8-ary constellations including 8-ary quadrature amplitude modulation (8-QAM), 8-ary phase-shift keying (8-PSK), circular (4,4)-QAM, and circular (7,1)-QAM. The superiority of DFT-S-OFDM-DM using circular (7,1)-QAM constellation with interleaving-based constellation partitioning over other benchmark schemes has been successfully verified by both simulation and experimental results. It is shown by the experimental results that a significant distance extension of 44.6% is obtained by DFT-S-OFDM-DM using circular (7,1)-QAM constellation with interleaving-based constellation partitioning in comparison to DFT-S-OFDM with index modulation achieving the same spectral efficiency of 2.5 bits/s/Hz. It is also demonstrated that the proposed constellation design schemes are also generally applicable to the constellation with an arbitrary shape and an arbitrary size.