We present the results of searches for B decays to charmless final states involving varphi, f_{0}(980), and charged or neutral rho mesons. The data sample corresponds to 384x10;{6} BB[over ] pairs collected with the BABAR detector operating at the PEP-II asymmetric-energy e;{+}e;{-} collider at SLAC. We find no significant signals and determine the following 90% confidence level upper limits on the branching fractions, including systematic uncertainties: B(B;{0}-->varphivarphi)<2.0x10;{-7}, B(B;{+}-->varphirho;{+})<30x10;{-7}, B(B;{0}-->varphirho;{0})<3.3x10;{-7}, B[B;{0}-->varphif_{0}(980)]xB[f_{0}(980)-->pi;{+}pi;{-}]<3.8x10;{-7}, and B[B;{0}-->f_{0}(980)f_{0}(980)]xB[f_{0}(980)-->pi;{+}pi;{-}]xB[f_{0}(980)-->K;{+}K;{-}]<2.3x10;{-7}.