Summary: In the Thom’s approach to the classification of instabilities in one-dimensional classical systems every equilibrium is assigned a local minimum in one of the Arnold’s benchmark potentials \(V_{(k)}(x)=x^{k+1}+c_1x^{k-1}+\dots\). We claim that in quantum theory, due to the tunneling, the genuine catastrophes (in fact, abrupt “relocalizations” caused by a minor change of parameters) can occur when the number \(N\) of the sufficiently high barriers in the Arnold’s potential becomes larger than one. A systematic classification of the catastrophes is then offered using the variable mass term \(\hbar^2/(2\mu)\), odd exponents \(k=2N+1\) and symmetry assumption \(V_{(k)}(x)=V_{(k)}(-x)\). The goal is achieved via a symbolic-manipulation-based explicit reparametrization of the couplings \(c_j\). At the not too large \(N\), a surprisingly user-friendly recipe for a systematic determination of parameters of the catastrophes is obtained and discussed.