Unlike most civilian solid-fuel rockets, which use ammonium perchlorate/aluminium fuels with a rubber binder (known in the industry as ammonium perchlorate composite propellant, or APCP for short), newer ICBMs (notably the U.S. “Peacekeeper” - oh, the irony) and ICBM-derived launch vehicles (such as the Minotaur IV, which is literally a Peacekeeper with a satellite on top instead of a bunch of warheads) use HMX (which probably stands for “Her Majesty’s Explosive” - it was a British invention - although there are other possibilities), also known as octogen, as their fuel.1 HMX provides considerably better performance than APCP, which is important for an ICBM with a bunch of heavy nuclear warheads on top.
However, HMX (along with its close sibling RDX, or hexogen) is a high explosive, and not a particularly insensitive one at that (it isn’t quite sensitive enough to go off when hit by a hammer, like PETN is, but it comes close). Besides, the main way even insensitive explosives are insensitive is that they require both great pressure and great heat to detonate, which keeps the explosive from going boom if you stomp on it (pressure, but not heat) or set it on fire (heat, but not pressure), but the interior of a burning SRB has a great abundance of both; even if you manage to light it without immediate disaster, the temperature and pressure inside the motor cavity should quickly rise to levels sufficient to trigger a deflagration-to-detonation transition and blow shrapnel all over the next few counties. Indeed, the explosive hazard of HMX is the reason everyone but the missileers (and the people they fobbed off their ex-missiles on) has shied away from it as a rocket propellant, despite its better performance, and stuck with APCP for their SRBs.
How, then, do HMX-fuelled rockets not detonate?
1: More recently, they’ve developed hexanitrohexaazaisowurtzitane, known to the military as CL-20 (China Lake 20), which has an even higher performance than HMX, but a) is another high explosive, and b) even if they can keep it from blowing up, won’t be used anyway unless the U.S. develops another ICBM, which appears somewhat unlikely given the modern emphasis on arms - especially nuclear and nuclear-capable arms - reduction.