Strained-silicon/relaxed-silicon-germanium alloy (strained-Si/SiGe) heterostructures are the foundation of Group IV-element quantum electronics and quantum computation, but current materials quality limits the reliability and thus the achievable performance of devices. In comparison to conventional approaches, single-crystal SiGe nanomembranes are a promising alternative as substrates for the epitaxial growth of these heterostructures. Because the nanomembrane is truly a single crystal, in contrast to the conventional SiGe substrate made by compositionally grading SiGe grown on bulk Si, significant improvements in quantum electronic-device reliability may be expected with nanomembrane substrates. We compare lateral strain inhomogeneities and the local mosaic structure (crystalline tilt) in strained-Si/SiGe heterostructures that we grow on SiGe nanomembranes and on compositionally graded SiGe substrates, with micro-Raman mapping and nanodiffraction, respectively. Significant structural improvements are found using SiGe nanomembranes.