The biharmonic oscillator and the asymmetric linear well are two confining power-law-type potentials for which complete bound-state solutions are possible in both classical and quantum mechanics. We examine these problems in detail, beginning with studies of their trajectories in position and momentum space, evaluation of the classical probability densities for both "x" and "p", and calculation of the corresponding quantum-mechanical solutions which give [vertical bar][psi][subscript n](x)[vertical bar][squared] and [vertical bar][phi](p)[vertical bar][squared] for comparison to their classical counterparts in the classically allowed regions. We then focus on the behavior of [phi][subscript n](p) for large momenta, motivated by recent studies of the very direct connections between the continuity behavior of V(x) and the large-"p" limit of the momentum-space wavefunction. (Contains 8 figures.)