<p>New rotating dilaton black hole and black string solutions in three spacetime dimensions are obtained. The solutions are asymptotically flat and, they are exact in classical string theory. The black hole solutions have only a single horizon. Enclosed inside their horizons, they contain a curvature singularity. The black strings carry axion charges and have two horizons. Depending on the ratio of their inner and outer horizons radii, they may or may not contain a curvature singularity. When they contain a singularity, the singularity is either at or enclosed inside their inner horizons. We also show that a solution with a constant or asymptotically non-zero Kalb-Ramond field is equivalent to a solution with no Kalb-Ramond field and non-zero rotation at asymptotic infinity. In general, non-zero rotation allows negative mass solutions. To demonstrate, we give negative mass black hole solutions. We also discuss black hole and black string solutions with a curvature singularity at or beyond their outer or event horizons. We also present novel black hole solutions with a ring curvature singularity in between their inner and outer horizons.</p>