Let \(G\) be a complete metric group. A set \(X\subset G\) is said to be shy if there exists a probability measure \(m\) on \(G\) such that \(m(gXh)= 0\) for all \(g,h\in G\). It is called prevalent if the set \(G_s X\) is shy. Let \(H\) denote the group of all measure-preserving autohomeomorphisms of the \(n\)-dimensional cube \(I^n\) with the distance \(d\) defined as follows: \[ d(f, g)= \max\Biggl( \max_{x\in I^n}|f(x)- g(x)|,\;\max_{x\in I^n} |f^{- 1}(x)- g^{- 1}(x)|\Biggr). \] Let \(G\) be the additive group of real continuous functions, defined on the Cantor set, equipped with the Chebyshev metric.
The paper contains a series of open problems connected with the prevalence of (a) the subset of \(H\) consisting of ergodic mappings and the subset consisting of unstable mappings, and (b) the subset of \(G\) consisting of functions which avoid null sets and the subset consisting of so-called clever functions.