Precise nanometre-scale imaging of soft structures at room temperature poses a major challenge to any type of microscopy because fast thermal fluctuations lead to significant motion blur if the position of the structure is measured with insufficient bandwidth. Moreover, precise localization is also affected by optical heterogeneities, which lead to deformations in the imaged local geometry, the severity depending on the sample and its thickness. Here we introduce quantitative thermal noise imaging, a three-dimensional scanning probe technique, as a method for imaging soft, optically heterogeneous and porous matter with submicroscopic spatial resolution in aqueous solution. By imaging both individual microtubules and collagen fibrils in a network, we demonstrate that structures can be localized with a precision of ~10 nm and that their local dynamics can be quantified with 50 kHz bandwidth and subnanometre amplitudes. Furthermore, we show how image distortions caused by optically dense structures can be corrected for.