Single-photon spectroscopy of pulsed, high-intensity sources of hard X-rays - such as laser-generated plasmas - is often hampered by the pileup of several photons absorbed by the unsegmented, large-volume sensors routinely used for the detection of high-energy radiation. Detectors based on the Timepix chip, with a segmentation pitch of 55 μm and the possibility to be equipped with high-Z sensor chips, constitute an attractive alternative to commonly used passive solutions such as image plates. In this report, we present energy calibration and characterization measurements of such devices. The achievable energy resolution is comparable to that of scintillators for γ spectroscopy. Moreover, we also introduce a simple two-detector Compton polarimeter setup with a polarimeter quality of (98 ± 1)%. Finally, a proof-of-principle polarimetry experiment is discussed, where we studied the linear polarization of bremsstrahlung emitted by a laser-driven plasma and found an indication of the X-ray polarization direction depending on the polarization state of the incident laser pulse.