I don’t have the Handbook near me at the moment so I can’t look at the example you mention, but at https://vaughnclimenhaga.wordpress.com/2013/11/24/fubini-foiled/ I describe a construction of Milnor’s (see the Intelligencer article linked to there) that is my go-to example for an elementary instance of this phenomenon. It’s similar to Katok’s original example but not identical and avoids things like entropy.
EDIT: This answer by Pengfei to a related MO question gives a very simple example of a continuous foliation which has a holonomy map that is not absolutely continuous, essentially following the idea that Daniel Asimov described in the comments (put the middle-third Cantor set on one side of a square, a Cantor set of positive Lebesgue measure on the other side, and connect corresponding points by straight lines). However, this foliation does have the weaker absolute continuity property that if $A\subset [0,1]^2$ has positive (2-dimensional) Lebesgue measure, then for many leaves $W$ of the foliation, $A\cap W$ has positive (1-dimensional) Lebesgue measure. (Here "many leaves" means "a set of leaves whose union has positive 2-dimensional Lebesgue measure.) The foliation described in Milnor's article (and my blog post) fails to have even this weaker property.