We investigate the cores of fossil galaxy groups and clusters (`fossil systems') using archival Chandra data for a sample of 17 fossil systems. We determined the cool-core fraction for fossils via three observable diagnostics, the central cooling time, cuspiness, and concentration parameter. We quantified the dynamical state of the fossils by the X-ray peak/brightest cluster galaxy (BCG), and the X-ray peak/emission weighted centre separations. We studied the X-ray emission coincident with the BCG to detect the presence of potential thermal coronae. A deprojection analysis was performed for z < 0.05 fossils to obtain cooling time and entropy profiles, and to resolve subtle temperature structures. We investigated the Lx-T relation for fossils from the 400d catalogue to see if the scaling relation deviates from that of other groups. Most fossils are identified as cool-core objects via at least two cool-core diagnostics. All fossils have their dominant elliptical galaxy within 50 kpc of the X-ray peak, and most also have the emission weighted centre within that distance. We do not see clear indications of a X-ray corona associated with the BCG unlike that has been observed for some other objects. Fossils do not have universal temperature profiles, with some low-temperature objects lacking features that are expected for ostensibly relaxed objects with a cool-core. The entropy profiles of the z < 0.05 fossil systems can be well-described by a power law model, albeit with indices smaller than 1. The 400d fossils Lx-T relation shows indications of an elevated normalisation with respect to other groups, which seems to persist even after factoring in selection effects.