A magnetic field measurement system for measurement of weak magnetic field signals or a wearable assembly includes at least one magnetometer and a shield disposed around the magnetometer. The shield includes a first portion configured for positioning between the at least one magnetometer and a source of the weak magnetic field signals. The first portion is made of an amplitude-selective magnetic shield (ASMS) that preferentially passes magnetic fields having a magnetic field amplitude below a threshold (for example, 500 nT or less) and shields magnetic fields having a larger magnetic field amplitude.