Infrared dark clouds (IRDCs) are potential hosts of the elusive early phases of high mass star formation (HMSF). Here, we conduct an in-depth analysis of the fragmentation properties of a sample of 10 IRDCs, which have been highlighted as some of the best candidates to study HMSF within the Milky Way. To do so, we have obtained a set of large mosaics covering these IRDCs with Atacama Large Millimeter/submillimeter Array (ALMA) at Band 3 (or 3 mm). These observations have a high angular resolution (~3 arcsec; ~0.05 pc), and high continuum and spectral line sensitivity (~0.15 mJy/beam and ~0.2 K per 0.1 km/s^ channel at the N_2_H^+^ (1 - 0) transition). From the dust continuum emission, we identify 96 cores ranging from low to high mass (M = 3.4-50.9 M_{sun}_) that are gravitationally bound ({alpha}_vir_ = 0.3-1.3) and which would require magnetic field strengths of B = 0.3-1.0 mG to be in virial equilibrium. We combine these results with a homogenized catalogue of literature cores to recover the hierarchical structure within these clouds over four orders of magnitude in spatial scale (0.01-10 pc). Using supplementary observations at an even higher angular resolution, we find that the smallest fragments (<0.02 pc) within this hierarchy do not currently have the mass and-or the density required to form high-mass stars. None the less, the new ALMA observations presented in this paper have facilitated the identification of 19 (6 quiescent and 13 star-forming) cores that retain > 16 M_{sun}_ without further fragmentation. These high-mass cores contain trans-sonic non-thermal motions, are kinematically sub-virial, and require moderate magnetic field strengths for support against collapse. The identification of these potential sites of HMSF represents a key step in allowing us to test the predictions from high-mass star and cluster formation theories.