The rheology and microstructure of soft particle glasses during startup flow are studied using three-dimensional particle dynamics simulations at different particle volume fractions and shear rates. The behavior of transient stress depends on the applied shear rate. At high shear rates, soft particle glasses exhibit a static yield stress signaled by a stress overshoot followed by a relaxation to a steadystate value. The buildup of the stress is driven by an interplay between structural anisotropy due to an accumulation of particles along the compression axis and a depletion along the extension axis and a compression of particles that are soft and deformable. At low shear rates, the stress increase is monotonic and without any stress overshoot. The time scale at which structural anisotropy and the stress are maximum is correlated to the nonaffine dynamics of SPGs through the persistence time of shear-induced particle collisions and to the residence time of particles inside their transient cages. The static yield strain gamma(p) and the reduced static yield stress sigma(p)/sigma(y), where sigma(y) is the dynamic yield stress deduced from steady flow measurements, follow universal behaviors when correlated with the dimensional shear rate eta(s)<(gamma)over dot>/G(0), with eta(s) being the suspending fluid viscosity and G(0) the storage modulus, which expresses the competition between elastic restoring forces and viscous dissipation. Dense suspensions of thermosensitive core-shell colloids, star-like micelles, and poly(ethylene oxide)-protected silica particles follow the same universal curves, suggesting the generality of our results. (C) 2021 The Society of Rheology.