  The trace formula for the density of single-particle levels in the
two-dimensional radial power-law potentials, which nicely approximate the
radial dependence of the Woods-Saxon potential and quantum spectra in a bound
region, was derived by the improved stationary phase method. The specific
analytical results are obtained for the powers 4 and 6. The enhancement
phenomena near the bifurcations of periodic orbits are found to be significant
for the description of the fine shell structure. It is shown that the
semiclassical trace formulas for the shell corrections to the level density and
energy reproduce the quantum results with good accuracy through all the
bifurcation (symmetry breaking) catastrophe points, where the standard
stationary-phase method breaks down. Various limits (including the harmonic
oscillator and the spherical billiard) are obtained from the same analytical
trace formula.
