We have performed experiments on current-induced domain-wall motion sCIDWMd in the case of the domain walls sDWd trapped within the nanoscale constrictions in patterned NiFe structures. Direct observation of current-induced magnetization reversal was achieved and critical current densities jc were measured in the presence of easy-axis magnetic fields. The direction of CIDWM was found to be along the direction of the electron motion in absence of an applied magnetic field and in the direction of the field when in the presence of even relatively weak fields. Data for the field dependence of jc for both uniform and fast rising pulses suggest that the current, regardless of polarity, assists in the depinning of the DW. Only for the dc case does the data strongly reveal the influence of the electron pressure in promoting or hindering DW motion.