LM358 and you're asking if it's misbehaving?!
It's not oscillating, it's just stabilizing. Try a 4.7k pullup on the output, forcing it into class A operation. LM358 is NOTORIOUS with its class-B output stage having bad enough crossover distortion that, well, I for one don't even like using it for control purposes (error amp) if I can avoid it.
As for the application, I wonder how one would discriminate between real shockwave/sonic boom, and just loud crashes or sharp sounds. Is there anything distinctive about the waveform (leading vs. trailing edge)? Can a high explosive be reliably distinguished from a low explosive (e.g. firecracker), or the bullet crack and muzzle blast from a firearm (one or both of which may be supersonic!)? What about reflections and other filtering effects, how robust should it be to those? (Would it be in relatively free air, on top of a tower perhaps, or in a chase plane?) What kind of a microphone is that anyway, surely these are intense pressure levels and a regular mic will not suffice? (Most clip in the 90dBa range IIRC. For such loud signals, with a differentiator, I would have to guess you wouldn't even need a preamp, or much of one anyway, on a dynamic mic.) How compact does this thing need to be, could it just be a recorder instead, and later processing done to detect the desired wave shape? Would it be practical to run such analysis in real time with a modestly powerful (probably >30MHz) CPU, DSP, or SBC? (Differentiation is notoriously noise sensitive and a more nuanced signal processing method is almost certainly helpful. Even if just bandpass filtering, for example.)
Tim