The purpose of this research is to identify and characterize oscillatory thermal instability in organic peroxides that are used in vast quantities in industry and misused by terrorists. The explosive thermal decompositions of lauroyl peroxide, methyl ethyl ketone peroxide, and triacetone triperoxide are investigated computationally, using a continuous stirred tank reactor model and literature values of the kinetic and thermal parameters. Mathematical stability analysis is used to identify and track the oscillatory instability, which may be violent. In the mild oscillatory regime it is shown that, in principle, the oscillatory thermal signal may be used in microcalorimetry to detect and identify explosives. Stabilization of peroxide thermal decomposition via Endex coupling is investigated. It is usually assumed that initiation of explosive thermal decomposition occurs via classical (Semenov) ignition at a turning point or saddle-node bifurcation, but this work shows that oscillatory ignition is also characteristic of thermoreactive liquids and that Semenov theory and purely steady state analyses are inadequate for identifying a thermal hazard in such systems.