The collapse of weakly turbulent pre-stellar cores is a critical stage in the process of star formation. Being highly non-linear and stochastic, the outcome of collapse can only be explored theoretically by performing large ensembles of numerical simulations. Standard practice is to quantify the initial turbulent velocity field in a core in terms of the amount of turbulent energy (or some equivalent) and the exponent in the power spectrum (n ≡−dlog Pk/dlog k). In this paper, we present a numerical study of the influence of the details of the turbulent velocity field on the collapse of an isolated, weakly turbulent, low-mass pre-stellar core. We show that, as long as n ≳ 3 (as is usually assumed), a more critical parameter than n is the maximum wavelength in the turbulent velocity field, λMAX. This is because λMAX carries most of the turbulent energy, and thereby influences both the amount and the spatial coherence of the angular momentum in the core. We show that the formation of dense filaments during collapse depends critically on λMAX, and we explain this finding using a force balance analysis. We also show that the core has only a high probability of fragmenting if λMAX > RCORE/2 (where RCORE is the core radius), the dominant mode of fragmentation involves the formation and break-up of filaments and although small protostellar discs (with radius RDISC≲ 20 au) form routinely, more extended discs are rare. In turbulent, low-mass cores of the type we simulate here, the formation of large, fragmenting protostellar discs is suppressed by early fragmentation in the filaments.