We report an experimental investigation of the cutting of a thin brittle polymer sheet with a blunt tool. It was recently shown that the fracture path becomes oscillatory when the tool is much wider than the sheet thickness. Here we uncover two novel transitions from straight to oscillatory fracture by varying either the tilt angle of the tool or the speed of cutting, respectively. We denote these by angle and speed unzip instabilities and analyze them by quantifying both the dynamics of the crack tip and the final shapes of the fracture paths. Moreover, for the speed unzip instability, the straight crack lip obtained at low speeds exhibits out-of-plane buckling undulations (as opposed to being flat above the instability threshold) suggesting a transition from ductile to brittle fracture.