Matrix stiffness that is sensed by a cell or measured by a purely physical probe reflects the intrinsic elasticity of the matrix and also how thick or thin the matrix is. Here, mesenchymal stem cells (MSCs) and their nuclei spread in response to thickness-corrected matrix microelasticity, with increases in nuclear tension and nuclear stiffness resulting from increases in myosin-II and lamin-A,C. Linearity between the widely varying projected area of a cell and its nucleus across many matrices, timescales, and myosin-II activity levels indicates a constant ratio of nucleus-to-cell volume, despite MSCs' lineage plasticity. Nuclear envelope fluctuations are suppressed on the stiffest matrices, and fluctuation spectra reveal a high nuclear tension that matches trends from traction force microscopy and from increased lamin-A,C. Transcriptomes of many diverse tissues and MSCs further show that lamin-A,C's increase with tissue or matrix stiffness anti-correlates with lamin-B receptor (LBR), which contributes to lipid/sterol biosynthesis. Adipogenesis (a soft lineage) indeed increases LBR:lamin-A,C protein stoichiometry in MSCs versus osteogenesis (stiff). The two factors compete for lamin-B in response to matrix elasticity, knockdown, myosin-II inhibition, and even constricted migration that disrupts and segregates lamins in situ. Matrix stiffness-driven contractility thus tenses the nucleus to favor lamin-A,C accumulation and suppress soft tissue phenotypes.