Flexpad is an interactive system that combines a depth camera and a projector to transform sheets of plain paper or foam into flexible, highly deformable, and spatially aware handheld displays. We present a novel approach for tracking deformed surfaces from depth images in real time. It captures deformations in high detail, is very robust to occlusions created by the user's hands and fingers, and does not require any kind of markers or visible texture. As a result, the display is considerably more deformable than in previous work on flexible handheld displays, enabling novel applications that leverage the high expressiveness of detailed deformation. We illustrate these unique capabilities through three application examples: curved cross-cuts in volumetric images, deforming virtual paper characters, and slicing through time in videos. Results from two user studies show that our system is capable of detecting complex deformations and that users are able to perform them quickly and precisely.