  The interplay between structural and electronic degrees of freedom in complex
materials is the subject of extensive debate in physics and materials science.
Particularly interesting questions pertain to the nature and extent of
pre-transitional short-range order in diverse systems ranging from shape-memory
alloys to unconventional superconductors, and how this microstructure affects
macroscopic properties. Here we use neutron and X-ray scattering to uncover
universal structural fluctuations in La$_{2-x}$Sr$_x$CuO$_4$ and
Tl$_2$Ba$_2$CuO$_{6+{\delta}}$, two cuprate superconductors with distinct point
disorder effects and optimal superconducting transition temperatures. The
fluctuations are present in wide doping and temperature ranges, including
compositions that maintain high average structural symmetry, and they exhibit
unusual, yet simple scaling laws. We relate this behavior to pre-transitional
phenomena in a broad class of systems with martensitic transitions, and argue
that it can be understood as a rare-region effect caused by intrinsic, doping-
and compound-independent nanoscale inhomogeneity. We also uncover remarkable
parallels with superconducting fluctuations, which indicates that the
underlying inhomogeneity plays a pivotal role in cuprate physics.
