The isospin asymmetries of B -> K-()mu(+)mu(-) decays and the partial branching fractions of B-0 -> K-0 mu(+)mu(-) and B+ -> K+mu(+)mu(-) are measured as a function of the di-muon mass squared q(2) using an integrated luminosity of 1.0 fb(-1) collected with the LHCb detector. The B -> K mu(+)mu(-) isospin asymmetry integrated over q(2) is negative, deviating from zero with over 4 sigma significance. The B -> Kmu(+)mu(-) decay measurements are consistent with the Standard Model prediction of negligible isospin asymmetry. The observation of the decay B0 -> K-s(0)mu(+)mu(-) is reported with 5.7 sigma significance. Assuming that the branching fraction of B-0 -> K-0 mu(+)mu(-) is twice that of B-0 -> K-s(0)mu(+)mu(-), the branching fractions of B-0 -> K-0 mu(+)mu(-) and B -> K(+)mu(+)mu(-) it are found to be (0.31(-0.06)(+0.07)) x 10(-6) and (1.16 +/- 0.19) x 10(-6), respectively.