Although AlInN is originally a wurtzite structure, zincblende and rocksalt are other potential phases. It will be interesting to have a comparative study of the physical properties of this compound in various phases. A DFT-based study of wurtzite, zincblende and rocksalt phases of AlInN alloys is carried out. Structural (lattice parameter, bulk modulus) and electronic properties (energy band gap, and electron effective mass) of the Al1 - xInxN alloys are investigated, at ambient pressure, throughout the whole range of indium contents for all considered phases. High pressure effects on the studied parameters are also examined, with the phase transition pressures computed for different values of In concentrations, and compared with available data. Structural density functional calculations are performed with Perdew-Burke-Ernzerhof gradient-corrected functional for solids (PBEsol), while electronic structure is computed with the modified Becke-Johnson (TB-mBJ) potential exchange to ensure a better accuracy of calculated the band gaps. Alloy randomness is taken into account using a special quasi-random structure.