The crucial effect of compressibility of rods on their instability is novelly demonstrated via singularity theory. It is shown that the critical load of compressible rod is always greater than the one of the Euler rod, and a subcritical pitchfork bifurcation, which cannot occur for the Euler rod, may occur for a compressible rod. A whole bifurcation diagram of compressible rods is as follows :when the original slenderness ratio of a compressible rod, S0, is smaller than (1 + ν/3)3√3π/2, the rod does not buckle; when S0 ∈ [ (1+ ν/3)3√3π/2, ( 1 + ν/5)√5π/4),the rod may undergo a subcritical pitchfork bifurcation and a collapse may occur; when S0 ∈ [(1 +ν/5)5 √5π/4, +∞ ), the rod may undergo a supercritical pitchfork bifurcation. The deformation of cross section of rods causes a little shift of bifurcation points towards to the one corresponding to larger slenderness ratio.