Consider the simple linear model y_j=β₀+β₁x_j+σz₀ (j=1,…,n) where z₁,z₂,…,z_nare i.i.d.errors with pdf [π(1+z²)]^-1, -∞<z<∞. The paper considers the joint estimation and test of hypothesis regarding the parameter (β₀,β₁,σ)' based on a few selected “regression quantiles” introduced by Koenker and Bassett (KB) (1978). The question of “optimum spacings” is addressed to the two problems. Further, the estimation of conditional regression quantiles is addressed along with the related optimum spacings. In every case, the optimum spacings are independent of the design matrix.