The recently suggested regular-type triangular quadrature amplitude modulation (TQAM) provides considerable power gain over square quadrature amplitude modulation (SQAM) at the expense of a slight increase in detection complexity. However, the power gain of the TQAM is limited due to the constraint that signal points should be regularly located at the vertexes of contiguous equilateral triangles. In this paper, we investigate two irregular (optimum and suboptimum) TQAMs where signal points are irregularly distributed while preserving the equilateral triangular lattice, and calculate achievable power gains of the proposed constellations. We also address optimum and suboptimum bit stream mapping methods and suggest a simple and optimum detection method for the constellations to be meaningful in practical implementation, and present analytical and simulation results. The proposed constellations can provide the asymptotic power gains of 0.825dB and 0.245dB over SQAM and regular TQAM, respectively.