Singularity of maps of several variables and a problem of Mycielski concerning prevalent homeomorphisms

S. Banach pointed out that the graph of generic (in sense Baire category) element $\text{Homeo}([0,1])$ has length $2$. J. Mycielski asked if measure theoretic dual holds, i.e., all but Haar null many Christensen) elements have We answer this question in affirmative. call $f \in \text{Homeo}([0,1]^d)$ singular it takes a suitable set full to nullset, and strongly is almost everywhere differentiable with derivative matrix. Since \text{Homeo}([0,1])$ $2$ iff $f$ singular, following results are higher dimensional analogues Banach's observation our solution Mycielski's problem. show for $d \ge 2$ $\text{Homeo}([0,1]^d)$ infinite $d$-dimensional Hausdorff measure, contrasting above result Banach. The remains open, we not null. singular. also every form so called ambivalent (neither null, nor co-Haar null). Finally, order clarify situation, investigate various possible definitions singularity maps several variables, explore connections between them.