The conformal anomaly indicates the breaking of conformal symmetry (angle-preserving transformations) in quantum theory by quantum fluctuations, and is a close cousin of the gravitational anomaly. We show that the conformal anomaly controls the variance of the local magnetization Mloc at finite temperatures in spin chains and spin ladders. This effect is predicted to appear at constant and variable temperature across the sample. The change of Mloc induced by the conformal anomaly is of the order of 3-5% of angular momentum 2 at 1 K for (C7H10N)2CuBr4 (DIMPY) or copper pyrazine nitrate (CuPzN) and increases linearly with temperature. Further, for a temperature gradient of 10% across the sample, the time relaxation of the nonequilibrium Mloc is of the order of nanoseconds. Thus, we believe that experimental techniques such as neutron scattering, nuclear magnetic resonance (NMR), spin noise, and ultrafast laser pumping should determine the presence of the conformal anomaly. Therefore, we outline a method to detect the conformal anomaly in spin observables of strongly interacting low-dimensional magnets.