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