We present the results of searches for decays of B mesons to final states with a b1 meson and a neutral pion or kaon. The data, collected with the BABAR detector at the Stanford Linear Accelerator Center, represent 465×106 BB̄ pairs produced in e+e- annihilation. The results for the branching fractions are, in units of 10-6, B(B+→b1+K0)=9.6±1.7±0.9, B(B0→b10K0)=5.1±1.8±0.5 (<7.8), B(B+→b1+π0)=1. 8±0.9±0.2 (<3.3), and B(B0→b10π0)=0.4±0. 8±0.2 (<1.9), with the assumption that B(b1→ωπ)=1. We also measure the charge asymmetry Ach(B+→b1+K0)=-0.03±0.15±0. 02. The first error quoted is statistical, the second systematic, and the upper limits in parentheses indicate the 90% confidence level. © 2008 The American Physical Society.