Phase-field simulations are performed to explore the thermal solidification of a pure melt in three-dimensional capillaries. Motivated by our previous work for isotropic or slightly anisotropic materials, we focus here on the more general case of anisotropic materials. Different channel cross sections are compared (square, hexagonal, circular) to reveal the influence of geometry and the effects of a competition between the crystal and the channel symmetries. In particular, a compass effect toward growth directions favored by the surface energy is identified. At given undercooling and anisotropy, the simulations generally show the coexistence of several growth modes. The relative stability of these growth modes is tested by submitting them to a strong spatiotemporal noise for a short time, which reveals a subtle hierarchy between them. Similarities and differences with experimental growth modes in confined geometry are discussed qualitatively.