We present the results of searches for decays of B mesons to final states with a b1 meson and a charged pion or kaon. The data, collected with the BABAR detector at the Stanford Linear Accelerator Center, represent 382×106 BB̅ pairs produced in e+e- annihilation. The results for the branching fractions are, in units of 10-6, B(B+→b10π+)=6.7±1.7±1.0, B(B+→b10K+)=9.1±1.7±1.0, B(B0→b1∓π±)=10.9±1.2±0.9, and B(B0→b1-K+)=7.4±1.0±1.0, with the assumption that B(b1→ωπ)=1. We also measure charge and flavor asymmetries Ach(B+→b10π+)=0.05±0.16±0.02, Ach(B+→b10K+)=-0.46±0.20±0.02, Ach(B0→b1∓π±)=-0.05±0.10±0.02, C(B0→b1∓π±)=-0.22±0.23±0.05, ΔC(B0→b1∓π±)=-1.04±0.23±0.08, and Ach(B0→b1-K+)=-0.07±0.12±0.02. The first error quoted is statistical, and the second systematic.