In this paper, we consider the structure of the AdS(3) vacua in R-3 expansion of new massive gravity (R-3-NMG). We obtain the degeneracies of the AdS(3) vacua at several points of the parametric space. Additionally, following a specific analysis we show that AdS(3) wave solutions are present. Using these wave solutions, we single out two special points of the parametric space for which logarithmic terms appear in the solutions. The first one is a point at which the effective mass of the wave profile, which is interpreted as a scalar mode, completely saturates the Breitenlohner-Freedman bound of the AdS(3) space in which the wave is propagating. The second special point is a point at which the central charge of the theory vanishes. Furthermore, we investigate the possibility of asymptotically Lifshitz black hole solutions to be present in the three-dimensional R-3-NMG. We derive analytically the Lifshitz vacua considering specific relations between the mass parameters of R-3-NMG. A certain polynomial equation arises at the first special point where solutions with logarithmic falloff in the AdS(3) space appear. Solving this polynomial equation, we obtain the values of the dynamical exponent z which correspond to possible asymptotically Lifshitz black hole solutions. However, it is shown that asymptotically Lifshitz black hole solutions do not exist in the three-dimensional R-3-NMG for a specific ansatz of the black hole metric.