We use a computationally efficient phase-field formulation [B. Echebarria et al., Phys. Rev. E 70, 061604 (2004)] to investigate the origin and dynamics of sidebranching in directional solidification for realistic parameters of a dilute alloy previously studied experimentally [M. Gorgelin and A. Pocheau, Phys. Rev. E 57, 3189 (1998)]. Sidebranching is found to result either from noise amplification or from deterministic oscillations that exist both in two dimensions and in a three-dimensional thin-sample geometry. The oscillatory branch of growth solutions bifurcates subcritically from the main steady-state branch of solutions and exists over a finite range of large array spacings. In contrast, noise-induced sidebranching is associated with a smooth transition where the sidebranching amplitude increases exponentially with spacing up to nonlinear saturation due to the overlap of diffusion fields from neighboring cells, as observed experimentally. In the latter case where sidebranching is noise-induced, we find that increasing the externally imposed thermal gradient reduces the onset velocity and wavelength of sidebranching, as also observed experimentally. We show that this counterintuitive effect is due to tip blunting with increasing thermal gradient that promotes noise amplification in the tip region.