We present the implementation of the radiative corrections of the Higgs sector in three public computer codes for the evaluation of the particle spectrum in the Minimal Supersymmetric Standard Model, Softsusy, Spheno and SuSpect. We incorporate the full one-loop corrections to the Higgs boson masses and the electroweak symmetry breaking conditions, as well as the two-loop corrections controlled by the strong gauge coupling and the Yukawa couplings of the third generation fermions. We include also the corrections controlled by the tau Yukawa coupling that we derived for completeness. The computation is consistently performed in the DRbar renormalisation scheme. In a selected number of MSSM scenarios, we study the effect of these corrections and analyse the impact of some higher order effects. By considering the renormalisation scheme and scale dependence, and the effect of the approximation of zero external momentum in the two-loop corrections, we estimate the theoretical uncertainty on the lighter Higgs boson mass to be 3 to 5 GeV. The uncertainty on Mh due to the experimental error in the measurement of the SM input parameters is approximately of the same size. Finally, we discuss the phenomenological consequences, using the latest value of the top quark mass. We find, in particular, that the most conservative upper bound on the lighter Higgs boson mass in the general MSSM is Mh < 152 GeV and that there is no lower bound on the parameter tan(beta) from non-observation of the MSSM Higgs bosons at LEP2.