Rectifiable sets are widely studied in geometric measure theory and in several areas of application. Even more interesting is the higher order version of such sets which has been of current investigation in the Euclidean setting. In this talk, we shall consider such inquiry in a metric setting - the Heisenberg groups. In particular, we give a characterization of higher order ($C^{1,\alpha}$-) rectifiable sets of low codimension in the spirit of almost every where existence of \textit{approximate tangent paraboloids}.